Structure of Free Semigroupoid Algebras

STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS

KENNETH R. DAVIDSON, ADAM DOR-ON, AND BOYU LI

Abstract. A free semigroupoid algebra is the wot-closure of the algebra generated by a TCK family of a graph. We obtain a structure theory for these algebras analogous to that of free semigroup algebra. We clar- ify the role of absolute continuity and wandering vectors. These results are applied to obtain a Lebesgue-von Neumann-Wold decomposition of TCK families, along with reflexivity, a Kaplansky density theorem and classification for free semigroupoid algebras. Several classes of examples are discussed and developed, including self-adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence.

Contents 1 Introduction1 2 Preliminaries9 3 Structure of the left regular algebra 13 4 Wold decomposition 16 5 Single vertex and cycle algebras 20 6 The structure theorem 25 7 Absolute continuity and wandering vectors 33 8 Lebesgue-von Neumann-Wold decomposition 40 9 Applications 43 10 Self-adjoint free semigroupoid algebras 49 11 Atomic representations 54 References 61

1. Introduction There has been a lot of interest in C*-algebras of directed graphs and their representations [47]. There have also been some studies on the norm closed and wot-closed algebras generated by graphs [40, 13, 41, 28]. In

2010 Mathematics Subject Classification. Primary: 47L80, 47L55, 47L40. Secondary: 46L05. Key words and phrases. graph algebra, free semigroupoid algebra, road coloring theorem, absolute continuity, wandering vectors, Lebesgue decomposition, operator algebras, Cuntz-Krieger algebras. The first author was partially supported by a grant from NSERC. The second author was partially supported by an Ontario Trillium Scholarship. 2 K.R. DAVIDSON, A. DOR-ON, AND B. LI particular, the papers of Muhly and Solel were heavily influenced by work of the first author with Pitts and others [19, 17, 14, 16] on free semigroup algebras. Here we are able to completely generalize the structure theory for free semigroup algebras to the directed graph context. One reason that C*-algebraist may be interested in this work is that the invariants for these wot-closed nonself-adjoint operator algebras (called free semigroupoid algebras) are also invariants for the Toeplitz-Cuntz-Kreiger family that determines the algebra. As such, it provides some interesting unitary invariants for representations of graph algebras. By the work of Glimm [30] (see [20]) we know that for any non-type I C*-algebra, there is no countable family of Borel functions which distin- guishes its irreducible representations up to unitary equivalence. In short, determining unitary equivalence of representations of non-type I C*-algebras is an intractable question in general. Two of the simplest examples of a non-type I C*-algebra are the Toeplitz- Cuntz and Cuntz algebras Tn and On respectively. There are several good reasons to want to distinguish their representations up to unitary equivalence. One of these lies in the work of Bratteli and Jorgensen [7,8,9] where various families of representations of On are considered, and are used to generate wavelet bases on self-similar sets, and to investigate endomorphisms of B(H). The theme in these results is to restrict to subclasses of representations, so as to parametrize them up to unitary equivalence. This has been done successfully by Davidson and Pitts [19] for atomic representations, by Davidson, Kribs and Shpigel [15] for finitely correlated representations, and by Dutkay and Jorgensen [23] for monic representations. Extending these results to subclasses of representations of classical Cuntz- Krieger algebras has become important, as can be seen in the work of Mar- colli and Paolucci [38] and the work of Bezuglyi and Jorgensen [5]. In these works, representations of Cuntz-Krieger algebras associated to semi- branching function systems are investigated, where monic representations are classified up to unitary equivalence, wavelets are constructed and Haus- dorff dimension is computed for self-similar sets. In fact, in recent years this has been pushed further by Farsi, Gillaspy, Kang and Packer [24] and together with Jorgensen [25] to representations of higher-rank graphs, where previous results were extended and expanded. For instance, building on work in [2] and [26], KMS states were realized on higher-rank graph C*- algebras, and connections with Hausdorff dimensions of associated spaces were made. A different way of addressing the problem of unitary equivalence, without restricting to a subclass, is by weakening the invariant. For representations of Tn, this is done through the work on free semigroup algebras. Every representation of Tn gives rise to a free semigroup algebra by taking the weak-∗ closed algebra generated by the row-isometry that defines the representation. When two representations are unitarily equivalent, their free STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 3 semigroup algebras are completely isometrically and weak-∗ homeomorphically isomorphic. Hence, one tries to determine isomorphism classes of free semigroup algebras instead. The theory of free semigroup algebras originates from the work of Popescu [43, 44, 45, 46], and Popescu and Arias [3] where the non-commutative Toeplitz algebra was introduced, and used to establish a robust dilation theory for row contractions. Popescu’s non-commutative Toeplitz algebra Ln is then just the free semigroup algebra generated by creation operators on full Fock space Fn on n symbols. In [3], Popescu and Arias describe the invariant subspaces and prove reflexivity of Popescu’s non-commutative Toeplitz algebra Ln in B(Fn). In a sequence of papers, Davidson and Pitts [19, 17, 18] introduce and study properties of Popescu’s non-commutative analytic Toeplitz algebra. They establish its hyperreflexivity, define free semigroup algebras and classify atomic free semigroup algebras. One turning point in the theory was the structure theorem of Davidson, Katsoulis and Pitts [15]. The structure theorem for free semigroup algebras allows for a very tractable 2 × 2 block structure where the (1, 1) corner is a von Neumann algebra, the (1, 2) corner is 0 and the (2, 2) corner is analytic in the sense that it is completely isometrically and weak-∗ homeomorphically isomorphic to Ln. This allowed Davidson, Katsoulis and Pitts to lift many results known for Ln to general free semigroup algebras. A serious roadblock in the theory, identified in [15], was the existence of wandering vectors for analytic free semigroup algebras. In [15, 16] it was shown that many spatial results that hold for Ln will hold for general free semigroup algebras if this roadblock was removed. This wandering vectors roadblock was eventually removed in the work of Kennedy [35], leading to a very satisfactory structure and decomposition theory for free semigroup algebras [36, 29]. These structure and decomposition results can then be applied to distinguish irreducible representations of On. For instance, we may classify them into analytic, von Neumann and dilation types, all of which are preserved under unitary equivalence. Representations obtained from dilating finite dimensional row contractions, known as finitely correlated representations, were classified by Davidson, Kribs and Shpigel [15]. They show that unitary equivalence of such representations reduces to unitary equivalence for associated (reduced) row contractions on a finite dimensional space, a problem in linear algebra. For a directed graph G = (V,E) with countably many edges E and vertices V , each edge e ∈ E has a source s(e) and range r(e) in V . We denote by + F (G) the collection of all paths of finite length in G, called the path space of G, or the free semigroupoid of G. 2 + Let HG ∶= ` (F (G)) be the Hilbert space with an orthonormal basis + {ξλ ∶ λ ∈ F (G)}. For each vertex v ∈ V and edge e ∈ E we may define 4 K.R. DAVIDSON, A. DOR-ON, AND B. LI projections and partial isometries given by ⎧ ⎧ ⎪ξ if r(µ) = v ⎪ξ if r(µ) = s(e) L (ξ ) = ⎨ µ and L (ξ ) = ⎨ eµ v µ ⎪ ( ) ≠ e µ ⎪ ( ) ≠ ( ) ⎩⎪0 if r µ v ⎩⎪0 if r µ s e and ⎧ ⎧ ⎪ξ if s(µ) = v ⎪ξ if s(µ) = r(e) R (ξ ) = ⎨ µ and R (ξ ) = ⎨ µe v µ ⎪ ( ) ≠ e µ ⎪ ( ) ≠ ( ) ⎩⎪0 if s µ v ⎩⎪0 if s µ r e The left and right free semigroupoid algebras given by G are wot LG ∶= Alg {Lv,Le ∶ v ∈ V, e ∈ E} and wot RG ∶= Alg {Rv,Re ∶ v ∈ V, e ∈ E}. LG can be thought of as the analogue of the non-commutative Toeplitz algebra for arbitrary directed graph, where Ln corresponds to the graph on a single vertex with n loops. In [37, 32, 31, 34] many of the results of Popescu and Arias, and of Davidson and Pitts on Ln were extended and expanded to LG for arbitrary graphs. While in these papers LG is called a free semigroupoid algebra, we will expand that definition here to be analogous to free semigroup algebras. The family L = (Lv,Le) is an example of a family satisfying the Toeplitz- Cuntz-Krieger relations. More precisely, we say that a family S = (Sv,Se) ∶= {Sv,Se ∶ v ∈ V, e ∈ E} of operators on a Hilbert space H is a Toeplitz-Cuntz- Krieger (TCK) family if

(P) {Sv ∶ v ∈ V } is a set of pairwise orthogonal projections; ∗ (IS) Se Se = Ss(e), for every e ∈ E ; ∗ −1 (TCK) ∑e∈F SeSe ≤ Sv for every finite subset F ⊆ r (v). The universal C*-algebra T (G) generated by families S satisfying the above relations is called the Toeplitz-Cuntz-Krieger algebra. If a Toeplitz-Cuntz- Krieger family S = (Sv,Se) satisfies the additional relation ∗ −1 (CK) ∑r(e)=v SeSe = Sv, for every v ∈ V with 0 < Sr (v)S < ∞, then we say that S a Cuntz-Krieger (CK) family. The universal such family generates the well-known graph C*-algebra O(G) associated to G, which is a quotient of T (G). Since we are dealing with wot-closed algebras, we will say that a CK family S is a fully coisometric if it satisfies the additional property ∗ (F) sot–∑r(e)=v SeSe = Sv for every v ∈ V . We will also say that a TCK family S is non-degenerate if

(ND) sot–∑v∈V Sv = I. It basically follows from the universal property of T (G) that ∗-representations of T (G) are in bijective correspondence with TCK families, where one identiﬁes a representation by the image of the canonical generators. As such, one can talk interchangeably about TCK families and ∗-representations of STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 5

T (G). These representations which factor through the quotient to O(G) are precisely those associated to CK families. Throughout the paper, we will denote the representation of T (G) associated to a TCK family S by πS. By a combination of a Wold type decomposition theorem (such as Theo- rem 4.2 or [32, Theorem 2.7]) and the gauge invariant uniqueness theorem for O(G), it follows that T (G) is ∗-isomorphic to C∗(L) via a map that sends generators to generators. Hence, we will always identify these algebras. We recommend [47] for more on graph C*-algebras and the gauge invariant uniqueness theorem for O(G).

Deﬁnition 1.1. Let G be a countable directed graph, and let S = (Sv,Se) be a non-degenerate TCK family for G acting on a Hilbert space H. The wot closed algebra S generated by S is called a free semigroupoid algebra of G.

An important class of examples for free semigroupoid algebras for a graph G are finitely correlated algebras that arise from the work of [28]. A free semigroupoid algebra S ⊆ B(H) is called finitely correlated if there exists a finite dimensional subspace V ⊆ H such that V is S∗-invariant, and cyclic for S. Finitely corrected representations of free semigroup algebras were first studied in [15] as an important class of examples, and results on them were extended to representations of product systems of C*-correspondences over k N by Fuller [28] (See also [6] in this context). The following is obtained as a special case of [28, Theorem 2.27] when the C*-correspondence arises from a directed graph. We denote by LG,v the restriction of each operator + in L to the reducing subspace HG,v ∶= {ξλ ∶ λ ∈ F (G), s(λ) = v} of basis vectors associated to paths emanating from v. We will say that that a free semigroupoid algebra is left-regular type if up to unitary equivalence it is (αv) generated by a a TCK family of the form ⊕v∈V LG,v for some multiplicities αv (some of which could be zero).

Theorem 1.2 (Fuller). Let S be the free semigroupoid algebra for a fully coisometric CK family S for a graph G. Assume that S is ﬁnitely correlated with ﬁnite dimensional cyclic coinvariant subspace V. Then

(1) There is a unique minimal cyclic coinvariant subspace Vˆ ⊆ V given by the span of all minimal coinvariant subspaces for S. (2) SSVˆ is a left-regular type free semigroupoid algebras. ˆ (3) A = ⊥PVˆ SSVˆ is a ﬁnite dimensional C*-algebra, where the Wedderburn ˆ ⊕ ˆ ⊕ decomposition is given by A ≃ ∑i Mdi on V ≃ ∑i Vi where each Vi is a minimal coinvariant subspace of dimension di. [V ] = S (4) S i is a minimal reducing subspace of S, and Si S S[Vi] is irreducible. The decomposition of Vˆ in item (3) yields a decomposition ⊕ of S into a ﬁnite direct sum of irreducible TCK families S ≃ ∑i Si 6 K.R. DAVIDSON, A. DOR-ON, AND B. LI

Hence, we obtain the following 2 × 2 block and ﬁnite sum decomposition

⎡ (α ) ⎤ ⊕ ⎢ M i 0 ⎥ ⎢ di ⎥ S = Q i . ⎢ d ⊕ (αv )⎥ i ⎢B( i , S[V ] ⊖ V ) L ⎥ ⎣ C i i ∑v∈V G,v( ) ⎦

Suppose S and T ﬁnitely-correlated TCK families with spaces VˆS and VˆT as in Theorem 1.2. By [28, Corollary 2.28] they are unitarily equivalent if and only if the operator families P ˆ SS ˆ and P ˆ TS ˆ are unitarily equivalent. VS VS VT VS When S is irreducible, and A = P SS = (Av,Ae), a computation that we VˆS VˆS leave to the reader shows that the multiplicities αv for v ∈ V can be recovered from the ranks of the projections {Av}v∈V , via the formula

αv = − rank(Av) + Q rank(As(e)). r(e)=v In this paper we will put Theorem 1.2 in the wider context of the structure for general free semigroupoid algebras. We next describe how this is accomplished. Deﬁnition 1.3. Let G be a directed graph and S be a free semigroupoid algebra generated by a TCK family S on H. A vector 0 ≠ x ∈ H is a { } wandering vector for S if Sµx µ∈F (G) are pairwise orthogonal. + Wandering vectors are useful for detecting invariant subspaces which are unitarily equivalent to left-regular free semigroupoid algebras. At the end of [41] Muhly and Solel ask if Kennedy’s wandering vector results can be extended beyond free semigroup algebras. The following is a simpliﬁed version of our structure theorem, and extends Kennedy’s wandering vector results in [35] to free semigroupoid algebras. Theorem 6.15 (Structure theorem). Let S be a free semigroupoid algebra on H generated by a TCK family S = (Sv,Se) for a directed graph G = (V,E). Let M = W ∗(S) be the von Neumann algebra generated by S. Then there is a projection P ∈ S such that (1) SP = MP so that P SP is self-adjoint. (2) If S is nonself-adjoint then P ≠ I. (3) P ⊥H is invariant for S. (4) With respect to H = P H ⊕P ⊥H, there is a 2×2 block decomposition

P MP 0 S = P ⊥MP SP ⊥

(5) When P ≠ I then SP ⊥ is completely isometrically and weak-∗ home- ′ omorphically isomorphic to LG for some induced subgraph G of G. ⊥ (6) P H is the closed linear span of′ wandering vectors for S. (7) If every vertex v ∈ V lies on a cycle, then P is the largest projection such that P SP is self-adjoint. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 7

Hence, we see that the situation described after Theorem 1.2 holds in greater generality, in line with the structure theorem from [14]. In an attempt to resolve the wandering vector problem, Davidson, Li and Pitts [16] introduced a notion of absolute continuity for free semigroup algebras. Muhly and Solel [41] extended this deﬁnition and generalized many of the results in [16] to isometric representations of W*-correspondences. In his second paper, Kennedy [36] related absolute continuity to wandering vectors, and proved that for free semigroup algebras on at least two generators, absolute continuity is equivalent to analyticity. As a consequence, he established a Lebesgue-von Neumann-Wold decomposition of representation of Tn. We say that a free semigroupoid algebra S generated by a TCK family S on H is absolutely continuous if the associated representation πS ∶ T (G) → B(H) has a weak-∗ continuous extension to LG. This deﬁnition was introduced in [41] for representations of W*-correspondences, and is motivated from operator theory where we ask for an extension of the analytic functional calculus of a contraction to an H∞ functional calculus. For every free semigroupoid algebra there is then a largest invariant subspace Vac on which it is absolutely continuous. We will say that S is regular ⊥ if the absolutely continuous part Vac coincides with P H where P is the structure projection from the structure Theorem 6.15. The following generalizes Kennedy’s characterization of absolute continuity. Theorem 7.10. Let S be a free semigroupoid algebra on H generated by a TCK family S = (Sv,Se) of a graph G. Then S is regular if and only if + (M) Whenever µ ∈ F (G) is a cycle such that Sµ is unitary on Ss(µ)H, the spectral measure of Sµ is either singular or dominates Lebesgue on T.

The above allows us to extend Kennedy’s decomposition theorem to representations of T (G). We will say that a free semigroupoid algebra S is of von Neumann type if it is a von Neumann algebra, and that it is of dilation type if P H is cyclic for S, and P ⊥H is cyclic for S∗ where P is the structure projection of Theorem 6.15. We have already seen in Theorem 1.2 that dilation type free semigroupoid algebras are certainly abundant, even when P H is ﬁnite dimensional. The following extends Kennedy’s Lebesgue decomposition theorem to regular TCK families. Theorem 8.5. Let G be a graph and S be a TCK family on H that generates a regular free semigroupoid algebra S. Then up to unitary equivalence we may decompose

S ≅ Sl ⊕ Sa ⊕ Ss ⊕ Sd where Sl is left-regular type, Sa is absolutely continuous, Ss is of von Neu- mann type and Sd is of dilation type. 8 K.R. DAVIDSON, A. DOR-ON, AND B. LI

Our results are applied to obtain a number of consequences including reflexivity of free semigroupoid algebras (see Theorem 9.1), a Kaplansky density theorem for regular free semigroupoid algebras (see Theorem 9.2) and an isomorphism type theorem for nonself-adjoint free semigroupoid algebras of transitive row-finite graphs (see Corollary 9.7). The first two results show that regular free semigroupoid algebras behave much like von Neumann algebras, while the third result shows that despite the above, when they are nonself-adjoint they still completely encode transitive row-finite graphs up to isomorphism. We conclude our paper by developing classes of examples of free semigroupoid algebras that illustrate our structure and decomposition theorems. In [14] it was asked if a free semigroup algebra on at least two generators can in fact be self-adjoint. In [48] Read was able to find a surprising example of two isometries Z1,Z2 with pairwise orthogonal ranges, such that the weak operator topology algebra generated by them is B(H). This points to the curious phenomenon that taking the wot closure of a nonself-adjoint algebra can suddenly make it into a von-Neumann algebra. Hence, it is natural to ask whether or not such self-adjoint examples occur for graphs that are on more than a single vertex. Suppose G = (V,E) is a finite, transitive and in-degree regular directed graph with in-degree d.A strong edge colouring of E is a function c ∶ E → {1, . . . , d} such that any two distinct edges with the same range have distinct colours. Every colouring of E then naturally induces a labeling of paths by words in the colours. We will say that a colouring as above is synchronizing + if for any (some) vertex v ∈ V , there is a word in colours γ ∈ Fd such that every path µ with the coloured label γ has source v. A famous conjecture of Adler and Weiss from the 70’s [1] asks if aperiodicity of the graph implies the existence of a synchronized colouring. In [49], Avraham Trahtman proved this conjecture, now called the road colouring theorem. We use the road colouring theorem to provide examples of self- adjoint free semigroupoid algebras for larger collection of graphs. Theorem 10.11. Let G be an aperiodic, in degree regular, transitive and finite directed graph. Then there exists a CK family S on K such that its associated free semigroupoid algebra S is B(K). Finally we discuss atomic representations of T (G), which are combinatorially defined analogues of TCK families. These are generalizations of those developed for representations of Tn in [19] by Davidson and Pitts, and are more general than permutative representations introduced by Brat- teli and Jorgensen in [8]. The associated TCK families have the property that there is an orthonormal basis on which the partial isometries associated to paths in the graph act by partial permutations, modulo scalars. We classify atomic representations up to unitary equivalence by decomposing them into left-regular, inductive and cycle types. For each type, we also characterize irreducibility of the representation. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 9

This paper contains 11 sections, including this introduction section. In Section2 we provide some preliminaries and notation to be used in the paper. In Section3 we explain how the left-regular free semigroupoid algebra changes by restricting to reducing subspaces and by changing the underlying graph. In Section4 we provide our variant of the Wold decomposition for TCK families and use it to characterize free semigorupoid algebras of certain acyclic graphs. In Section5 we reduce problems on wandering vectors to free semigroup algebras and characterize all free semigroupoid algebras of a single cycle. These results are used extensively in the rest of the paper. In Section6 we give a proof of the structure theorem. In Section7 we develop absolute continuity of free semigroupoid algebras and relate them to wandering vectors. In Section8 we generalize and extend Kennedy’s Lebesgue decomposition to free semigroupoid algebras. In Section9 we show free semigroupoid algebras are reﬂexive, provide a Kaplansky density theorem and prove isomorphism theorems for free semigroupoid algebras. In Section 10 we show that B(K) is a free semigroupoid algebra for transitive, in-degree regular, ﬁnite and aperiodic graphs. Finally, in Section 11 we classify atomic representations up to unitary equivalence by decomposing them into left-regular, inductive and cycle types.

2. Preliminaries A (countable) directed graph G = ((V, E, s, r) consists of a countable vertex set V and a countable edge set E, and maps s, r ∶ E → V . To each edge e ∈ E, we call s(e) its source and r(e) its range in V . One should think of e as an arrow or directed edge from s(e) to r(e). A finite path in G is a finite sequence µ = en . . . e1 where ei ∈ E and s(ei+1) = r(ei) for 1 ≤ i < n. We + denote by F (G) the collection of all paths of finite length in G, called the + path space of G, or the free semigroupoid of G. For each element µ ∈ F (G), define the source s(µ) = s(e1) and range r(µ) = r(en) in V . We can compose two paths by concatenation provided that the range and source match, i.e. if + µ, ν ∈ F (G), say ν = fm . . . f1, and s(µ) = r(ν), then µν = en . . . e1fm . . . f1. + This makes F (G) into a semigroupoid. Also, we denote by SµS ∶= n the + length of the path µ ∈ F (G). A vertex v is considered to be a path of length 0 with s(v) = r(v) = v. When S = (Sv,Se) is a TCK family, we define partial isometries for each + path µ ∈ F (G), where µ = en . . . e1 for ei ∈ E, by

Sµ = Sen ⋯Se1 . We will make use of the fact that the C*-algebra C∗(S) generated by a TCK family S = (Sv,Se) has the following description as a closed linear span, ∗ ∗ + C (S) = span{SµSν ∶ µ, ν ∈ F (G)}, as can be veriﬁed by the use of (IS). In other words, the ∗-algebra generated ∗ ∗ by S, which is norm dense in C (S), is the linear span of the terms SµSν + where µ, ν ∈ F (G). 10 K.R. DAVIDSON, A. DOR-ON, AND B. LI

Since we are interested in the wot-closed algebra, it is worth noting that property (F) is not readily identified at the norm closed level. Also if V is finite, then T (G) is unital, and non-degeneracy of the representation reduces to the statement that it is also unital. However if V is infinite, T (G) is not unital, and non-degeneracy is also not readily detected in the norm closed algebra T (G). For this reason, we will concentrate on TCK families rather than on representations of T (G). Corresponding statements about ∗-representations are usually left to the interested reader. The (left) tensor algebra determined by G is the norm closed algebra generated by L, + T+(G) ∶= span{Lµ ∶ µ ∈ F (G)}. From their universality, we can deduce that the algebras T+(G), T (G) and O(G) carry an additional grading by Z implemented by a point-norm continuous gauge action. However, a spatial description for this grading be more useful to us. The space HG is N-graded by components HG,n with + orthonormal basis {ξµ ∶ SµS = n, µ ∈ F (G)} for n ∈ N. We denote by En the projection onto HG,n. We may then define a unitary Wλ ∶ HG → HG for every λ ∈ T by specifying SµS ∗ Wλ(ξµ) = λ ξµ. Then α ∶ T → B(HG) given by αλ(T ) = WλTWλ becomes a group action satisfying αλ(Lv) = Lv and αλ(Le) = λ ⋅ Le for each v ∈ V and e ∈ E. Since T (G), T+(G) and LG are invariant under this action, we may look at α as an action on the algebras LG, T+(G) and T (G). In the first case it becomes a point-wot continuous action, and in the latter two it becomes a point-norm continuous action. For each of these algebras, this then enables the definition of Fourier coefficients, which are maps Φm from the algebra to itself. The formula is given for every m ∈ Z and T ∈ T (G) by −m Φm(T ) = S αλ(T )λ dλ. T The operator Φm(T ) maps each subspace HG,n into HG,n+m (possibly {0}). Then define the Cesàromeans by SmS Σk(T ) = Q 1 − Φm(T ). SmS

These statements are proven in [37]. Remark 2.1. Since there can be inﬁnitely many edges, even between two vertices, Σk(T ) may not belong to T+(G) when T ∈ LG. This can be reme- died using the net Σk(A)PF where PF = ∑v∈F Sv for ﬁnite subsets F ⊆ V . Observe that Φm(T )Sv = wot–∑SµS=m, s(µ)=v aµLµ. Because the isometries Lµ with SµS = m and s(µ) = v have pairwise orthogonal ranges and a common domain, we have 1~2 2 YT Y ≥ YΦm(T )Y ≥ [wot–Q aµLµ[ = Q SaµS . SµS=m SµS=m s(µ)=v s(µ)=v

It follows that the sequence (aµ) running over paths µ with SµS = m and s(µ) = v belongs to `2. Hence, the sum ∑ SµS=m aµLµ converges in norm. s(µ)=v Therefore Φm(T )PF belongs to T+(G); whence so does Σk(T )PF . Moreover Σk(T )PF converges in the wot topology to T ∈ LG as k → ∞ and F → V .

For a non-degenerate TCK family S = (Sv,Se) of a directed graph G acting on a Hilbert space H, we denoted by + S ∶= wot−Alg{Sµ ∶ µ ∈ F (G)} the free semigroupoid algebra generated by S inside B(H), which is unital. We will also deﬁne + AS ∶= Alg{Sµ ∶ µ ∈ F (G)} and ∗ + TS ∶= C ({Sµ ∶ µ ∈ F (G)}) to be the norm closed algebra and C*-algebra generated by S inside B(H), respectively. The underlying graph G will be clear from the context. An important way of constructing free semigroupoid algebras is through dilation of contractive G-families. Let A = (Av,Ae) be a family of operators on a Hilbert space H. We say that A is a contractive G-family if

(P) {Av ∶ v ∈ V } is a set of pairwise orthogonal projections; ∗ (C) Ae Ae ≤ As(e), for every e ∈ E ; ∗ −1 (TCK) ∑e∈F AeAe ≤ Av for every ﬁnite subset F ⊆ r (v). The family above is CK-coisometric if ∗ −1 (CK) ∑r(e)=v AeAe = Av, for every v ∈ V with 0 < Sr (v)S < ∞, and fully coisometric if we have ∗ (F) sot–∑v∈V ∑r(e)=v AeAe = Av for every v ∈ V . From the work of Muhly and Solel [39, Theorem 3.10], we know that completely contractive representations of T+(G) are in bijection with contractive G-families A as above. Hence, once again we can treat contractive G-families and completely contractive representations of T+(G) interchangeably. 12 K.R. DAVIDSON, A. DOR-ON, AND B. LI

Contractive G-families are a lot easier to construct than TCK families. For instance, there are graphs G with no TCK families on a finite dimensional Hilbert spaces, while on the other hand one can construct a contractive G-family for any graph on any finite dimensional space. In fact, free semigroupoid algebras generated by TCK dilations of contractive G-families on finite dimensional spaces are exactly the finitely correlated free semigroupoid algebras in Theorem 1.2. Given a free semigroupoid algebra S that is generated by a TCK family = ( ) H K ⊆ H = T S Sv,Se on , for any co-invariant subspace , define Aµ PKSµ K for µ ∈ F+(G). Then A is a contractive G-family. The graph-analogue of the Bunce, Frahzo, Popescu dilation theorem [27, 10, 43] gives us the converse. Indeed, if we start with a contractive G- family A = (Av,Ae) on a Hilbert space K, by [39, Theorem 3.3] we see that A always has a unique minimal dilation to a TCK family. More precisely, there is a TCK family S = (Sv,Se) for G on a Hilbert space H containing K = T ∈ ( ) as a co-invariant subspace such that Aµ PKSµ K for µ F+ G . In fact, S can be chosen minimal in the sense that K is cyclic for S, and any two minimal TCK dilations for A are unitarily equivalent. Moreover, by [39, Corollary 5.21] we have that S is CK when A is CK-coisometric, and is fully coisometric when A is fully coisometric. Hence, contractive G-families yield many interesting examples of TCK families via dilation. Let G = (V,E) be a directed graph. We call a subset F ⊆ V directed if whenever e ∈ E is such that s(e) ∈ F , then r(e) ∈ F . If F ⊆ V , let FG denote the smallest directed subset of V containing F in G. In general, we will denote by G[F ] ∶= G[FG] the subgraph induced from G on the smallest directed subset of V containing F . Given a TCK family S for G, denote by supp(S) the collection of vertices ∗ v ∈ V such that Pv ≠ 0. Since for each edge e ∈ E, we have Ss(e) = Se Se ∗ and SeSe ≤ Sr(e), we see that supp(S) is directed. Hence, if we take the subgraph G[S] ∶= G[supp(S)], we obtain a reduced TCK family −1 Sr ∶= {Sv,Se ∶ v ∈ supp(S), e ∈ s (supp(S))} for G[S] = (supp(S), s−1(supp(S))). This means that we may assume from the outset that Sv ≠ 0 for all v ∈ V by identifying S with a TCK family for an induced subgraph on a directed subset of vertices. We will say that a TCK family S for a graph G is fully supported if supp(S) = V . We say that a directed graph G is transitive if for any two vertices v, w ∈ V + we have a path µ ∈ F (G) with s(µ) = v and r(µ) = w. If G is transitive, then since supp(S) is directed, either supp(S) = ∅ or supp(S) = V . When the TCK family is non-degenerate, we must have supp(S) = V . Hence, any non-degenerate TCK family of a transitive graph is fully supported. Let µ = en . . . e1 be a cycle. We say µ is vertex-simple if s(ei) ≠ s(ej) for any 1 ≤ i ≠ j ≤ n. We say µ has an entry if there is an edge f ∈ E with r(f) = s(ei) for some 1 ≤ i ≤ n and f ≠ ej for all 1 ≤ j ≤ n. We will also say that a µ is irreducible if s(ei) ≠ s(e1) for all 1 ≤ i ≤ n. Note that an STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 13 irreducible cycle may fail to be vertex-simple, as it may visit other vertices, except for its initial vertex, along the way.

3. Structure of the left regular algebra In this section, we give some basic results about the structure of the left regular free semigroupoid algebra, relating them to their associated graph. For a general free semigroupoid algebra S generated by a TCK family S for a graph G on a Hilbert space H, if G is the union of countably many connected components (as an undirected graph), we have a corresponding decomposition of S. Indeed, suppose that G = " Gi is a union of undirected = ( ) H ∶= ( )H connected components Gi Vi,Ei . Let i sot–∑v∈Vi Sv . Then ⊕ ∞ H = ∑ Hi and SG = ∏ SGi is the ` -direct sum (direct product) of free semigroupoid algebras SGi for Gi acting on Hi. Hence, we may always assume that G = (V,E) is connected as an undi- ′ ′ ′ rected graph. If F,F ⊆ V are two subsets, and F ⊆ FG, we write F ≻ F ; i.e., F ≻ F ′ if F ′ is contained in the smallest directed subset containing F . This is a transitive relation on subsets of V . One can have F ≻ F ′ and F ′ ≻~ F . We will also write v ≻ w to mean that {v} ≻ {w}. Note that G is transitive precisely when v ≻ w for any two vertices v, w ∈ V . ⊕ Every path has a source, and hence HG ≅ ∑v∈V RvHG. For each subset F ⊆ V , we deﬁne ⊕ HG,F ∶= Q RvHG = span{ξλ ∶ s(λ) ∈ F }. v∈F In particular, we use

HG,v ∶= HG,{v} = RvHG = span{ξλ ∶ s(λ) = v} for v ∈ V.

Each HG,F is easily seen to be a reducing subspace for LG. For a subset

F ⊆ V , we also write LG,F = (LvSHG,F ,LeSHG,F ∶ v ∈ V, e ∈ E) for the family of operators obtained by restricting to HG,F and similarly LG,v ∶= LG,{v}. wot We deﬁne LG,F ∶= LGSHG,F and set LG,v ∶= LG,{v} and LG,v ∶= LG,{v} = wot LGSHG,v . We will see in Theorem 3.1 that the closure is unnecessary; i.e.,

LG,F = LGSHG,F . Theorem 3.1. Suppose that F ≻ F ′ are subsets of vertices of G. Then there is a complete quotient map ρF,F ∶ LG,F → LG,F given by the natural restriction map; i.e. LG,F is completely′ isometrically′ isomorphic and weak- ∗ homeomorphic to LG,F ~′ ker ρF,F . Moreover there is a canonical weak-∗ continuous completely isometric homomorphism′ jF ,F ∶ LG,F → LG,F which is a section of the quotient map. ′ ′ Likewise there is a canonical weak-∗ continuous completely isometric homomorphism jF ∶ LG,F → LG which is a section of the quotient map. ′ ′ ′ ′ Proof. Since F ≻ F , for each w ∈ F there is a path µw with s(µw) = vw ∈ F and r(µw) = w. There is a natural injection of HG,w into HG,vw given by Rµw , 14 K.R. DAVIDSON, A. DOR-ON, AND B. LI

+ where Rµw ξν = ξνµw for each ν ∈ F (G) with s(ν) = w. Thus, R = ⊕w∈F Rµw (α) ′ ′ provides a natural injection of HG,F into HG,F , where α = SF S. For each + λ ∈ F (G) with s(λ) = w, we then have′ R∗ L S R = L S . µw λ HG,s µw µw λ HG,w

( ) That is, the range of Rµw is an invariant subspace of HG,s(µw) such that the compression of L S to R H is canonically unitarily equivalent λ HG,s µw µw G,w to LλSHG,w . Hence, as this( ) occurs on each block separately, we get ∗( S )(α) = S R Lλ HG,F R Lλ HG,F .

′ ∗ (α) Therefore the map ρF,F ∶ LG,F → LG,F given by ρF,F (A) = R A R is a completely contractive homomorphism.′ ′ Since this map′ is the composition of an ampliation and a spatial map, it is weak-∗ continuous. This isometry R depends on the choice of the path µw, but the map ρF,F ′ is independent of this choice. To see this, suppose that µw is another set of′ ′ = ⊕ paths as above; and let R w∈F Rµw be the corresponding isometry. Then ′ ∶ → ′ ′ = the unitary map W Ran R Ran R given by W ξνµw ξνµw is readily seen ′ ′ + to satisfy WR = R and LλR = WLλR for each λ ∈ F (G). So′ the map ρF,F is the unique map satisfying ′ ( S ) = S ρF,F Lλ HG,F Lλ HG,F . ′ ′ Let A ∈ LG,F . Then YΣk(A)PH Y ≤ YAY, and the bounded net Σk(A)PH for k ≥ 1 and ﬁnite′ subsets H ⊆ V converges to A in the wot topology by + Remark 2.1. Observe that paths λ ∈ F (G) with non-zero coeﬃcients in the Fourier expansion of A all have source (and range) in the minimal directed ′ ′ subset FG of G containing F . However, being a polynomial, Σk(A)PH makes sense as an element of LG,F . Observe that when acting on HG,F , we ′ have Σk(A)ξµ = 0 if r(µ)∈ ~ FG. Indeed the subspace ′ K = span{ξµ ∈ HG,F ∶ r(µ) ∈ FG} is an invariant subspace of HG,F . We claim that K is contained in some multiple of HG,F and conversely HG,F is contained in some multiple of K. + Let M be the set′ of minimal length paths′ µ = en . . . e1 ∈ F (G) with s(µ) ∈ F ′ ′ and r(µ) ∈ FG. Hence, we have that r(ei)∈ ~ FG for i < n. We see that

K = Q ⊕ LG,r(µ)[ξµ] ≅ Q ⊕ RµHG,s(µ) ≅ Q ⊕ HG,r(µ). µ∈M µ∈M µ∈M The unitary equivalence here is again canonical and intertwines the action (α) of each Lλ. Therefore K is a reducing subspace of HG,F for α = SMS. On ′ ′ the other hand, since F ≻ F , K contains Rµw HG,vw for each′ w ∈ F . Thus (β) ′ K contains HG,F as an invariant subspace if β = SF S. The subspace HG,F′ ⊖K is spanned by the vectors ξν such that ν = fk . . . f1 ′ where r(fi)∈ ~ FG for 1 ≤ i ≤ k. It follows that Σk(A) vanishes on HG,v ⊖ K. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 15

Therefore we see that ( ) S ≅ S ⊕ ⊕ ( ) S Σk A PH HG,F 0 HG,v⊖K Q Σk A PH HG,r µ . µ∈M ( ) We have already observed above that YΣ (A)P S Y ≤ YΣ (A)P S Y k H HG,r µw k H HG,w ′ for w ∈ F . Therefore we deduce that ( ) Y ( ) S Y ≤ Y ( ) S Y ≤ Y ( ) S Y Σk A PH HG,F Σk A PH HG,F Σk A PH HG,F . Thus we have equality. We remark that the′ identical argument is valid for matrices in Mn(LG,F ). We use this to show′ that ρF,F is a complete quotient map. Take A in Mn(LG,F ). By the previous paragraph,′ we see that ′ (n) (n) YΣ (A)P Y = YΣ (A)P S n Y ≤ YAY. k H k H H G,F ( ) ′ By wot-compactness of the unit ball of Mn(LG,F ), there is a coﬁnal conver- gent subnet with limit B ∈ Mn(LG,F ) with YBY ≤ YAY. However the Fourier series of B is readily seen to coincide with the Fourier series of A. Indeed, using the setup of the previous paragraph, we see that ≅ S ⊕ ⊕ S B 0 HG,F ⊖K Q A HG,r µ . µ∈M ( ) In particular, the map jF ,F ∶ LG,F → LG,F given by jF ,F (A) = B is a completely isometric homomorphism′ ′ of LG,F into LG,F . The′ structure ○ = ′ of B makes it apparent that ρF,F jF ,F idLG,F . So this is the desired section. Since jF ,F is completely isometric,′ ′ it follows′ that ρF,F is a weak-∗ continuous complete′ quotient map. Since the composition of′ the section produced in the previous paragraph and the quotient is weak-∗ continuous as well, it follows that this isomorphism is a weak-∗ homeomorphism. The last statement of the theorem now follows from the fact that HG = ⊕ ′ ∑v∈V HG,v = HG,V , and V ≻ F . We obtain the following useful consequences.

Corollary 3.2. If G is a graph and F ⊆ V , then LGSHG,F = LG,F . Corollary 3.3. If G is a graph and F ⊆ V such that F ≻ w for all w ∈ V , then ρF ∶ LG → LG,F is a completely isometric isomorphism and weak-∗ homeomorphism. In particular when G is transitive, this occurs for any subset F = {v} for any vertex v ∈ V .

Corollary 3.4. Suppose that G has a collection {Fi}i∈I of subsets of V linearly ordered by I such that Fi ≻ Fj when i ≥ j, and for every w ∈ V , there is some i so that Fi ≻ w. Then LG is a wot-projective limit of the system

ρFi,Fj ∶ LG,Fi → LG,Fj for j ≤ i, and a wot-inductive limit of the system jFj ,Fi ∶ LG,Fj → LG,Fi for j ≤ i.

Proof. The maps ρFi ∶ LG → LG,Fi are the connecting maps which establish the projective limit. Likewise the maps jFi ∶ LG,Fi → LG are the connecting maps which establish the inductive limit. 16 K.R. DAVIDSON, A. DOR-ON, AND B. LI

4. Wold decomposition We prove a Wold-decomposition type theorem for general Toeplitz-Cuntz- Krieger families which is similar to [32, Section 2] and [21, Theorem 3.2]. A more general Wold decomposition in the setting of C*-correspondences can be found in [40, Corollary 2.15]. We show that once one removes all of the reducing subspaces determined by wandering spaces (the left regular part), what remains is a fully coisometric CK family for the largest sourceless subgraph G0 of G. Deﬁnition 4.1. Given a free semigroupoid algebra S generated by a TCK family S = (Sv,Se), we say that a non-trivial subspace W is a wandering subspace provided that {SµW ∶ µ ∈ F+(G)} are pairwise orthogonal. (Note that some of these subspaces may be {0}.) A wandering subspace determines the invariant subspace

S[W] = SW = Q ⊕ SλW.

λ∈FG + ⊕ We call a wandering subspace W total if W = ∑v∈V SvW. The support of W is the set supp(W) = {v ∈ V ∶ SvW ≠ {0}}. A vector 0 ≠ ξ is called wandering provided that Cξ is a wandering subspace, and supp(ξ) ∶= supp(Cξ). We say that W has full support if supp(W) = V . Note that a wandering subspace W is total exactly when ′ W = S[W] ⊖ Q ⊕ SλW =∶ W ,

λ∈FG + and that in general W′ is always a total wandering space with S[W′] = S[W]. When S is non-degenerate, every wandering subspace supported on a single vertex is total. For a TCK family S = (Sv,Se) on a Hilbert space H and a common invariant subspace K ⊆ H, we will write SSK ∶= (SvSK,SeSK). Furthermore, ′ for another TCK family T = (Tv,Te) on H , we will let

T ⊕ S ∶= (Tv ⊕ Sv,Te ⊕ Se) denote the TCK family direct sum on H ⊕ H′, and T(α) denotes the direct sum of T with itself α times. The support of a subspace W may not be directed, but it is easy to see that supp(SSS[W]) is the smallest directed set containing supp(W). In the following, any inﬁnite sum of orthogonal projections is evaluated as an sot-limit.

Theorem 4.2 (Wold decomposition I). Let S = (Sv,Se) be a non-degenerate TCK family for a graph G acting on a Hilbert space H generating the free semigroupoid algebra S. For any v ∈ V , deﬁne ∗ Wv ∶= Sv − Q SeSe H. e∈r 1(v) − STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 17

If Wv ≠ {0}, then Wv is a wandering subspace for the reducing subspace Hv = S[Wv] with supp(Wv) = {v}. In this case, let {ξv,i ∶ 0 ≤ i < αv} be an orthonormal basis for Wv, where αv = dim Wv. Then for each 0 ≤ i < αv, the set {Sµξv,i ∶ s(µ) = v} is an orthonormal basis for a reducing subspace Hv,i generated by the wandering vector ξv,i supported on v such that Sv,i ∶= SSHv,i is unitarily equivalent to L ; and H = ⊕ H . Furthermore, W = H ⊖ S S∗H = ⊕ W G,v v ∑i<αv v,i ∑e∈E e e ∑v∈V v ⊕ is a total wandering space and S[W] = ∑v∈V S[Wv] is the maximal reducing subspace such that SSS[W] is of left regular type. Thus, the TCK family S is unitarily equivalent to

(αv) T ⊕ ? LG,v v∈V where T = (Tv,Te) is a non-degenerate and fully coisometric CK family. This decomposition is unique up to unitary equivalence of T.

Proof. Note that Wv = SvWv, so clearly supp(Wv) = {v}. Suppose that f ∈ E. Then ⎧ ⎪S − S = 0 if r(f) = v, S − Q S S∗S = ⎨ f f v e e f ⎪ ( ) ≠ e∈r 1(v) ⎩⎪0 if r f v. − We will simultaneously show that Wv is wandering for all v ∈ V , and that Hv and Hw are orthogonal for all v ≠ w in V . Consider SµWv and SνWw where either v ≠ w or µ ≠ ν. We may suppose that SµS ≥ SνS. Write µ = ν′µ′ where Sν′S = ν. Also if Sµ′S ≥ 1, write µ′ = e′µ′′. Then we calculate for ξ, η ∈ H, ∗ ∗ aSµSv − Q SeSe ξ, SνSw − Q Sf Sf ηf e∈r 1(v) f∈r 1(v) − ∗ ∗ − ∗ = aSw − Q Sf Sf Sν Sν Sµ Sv − Q SeSe ξ, ηf 1 1 f∈r (v) ′ ′ e∈r (v) − ∗ − ∗ = δν,ν aSw − Q Sf Sf Sµ Sv − Q SeSe ξ, ηf 1 1 ′ f∈r (v) ′ e∈r (v) ⎪⎧0 if ν −≠ ν′, − ⎪ = ⎨0 if ν = ν′ = µ and v ≠ w since S S = 0, ⎪ w v ⎪ ′ ′ ′ ′′ ∗ ⎩0 if ν = ν and µ = e µ since Sw −∑f∈r 1(v)Sf Sf Se = 0. − ′ It follows that each Wv is wandering, and that Hv and Hw are othogonal for v ≠ w. Evidently Hv is an invariant subspace. The ﬁrst computation in the ∗ ∗ previous paragraph shows that if f ∈ E, then Sf Sv − ∑e∈r 1(v) SeSe = 0. ∗ ′ ∗ − Therefore Sf Wv = {0}. Moreover, if λ = eλ , then Sf SλW = δf,eSλ W. It ∗ ′ follows that Sf Hv ⊆ Hv. Therefore Hv is a reducing subspace. When Wv ≠ {0}, it is clear from the wandering property that the set {Sµξv,i ∶ s(µ) = v} is orthonormal and spans an invariant subspace Hv,i. ∗ ′ Again it is reducing because Sf ξv,i = 0 for all f ∈ E, and when µ = eµ , 18 K.R. DAVIDSON, A. DOR-ON, AND B. LI

∗ we have Sf Sµξv,i = δf,eSµ ξv,i. A similar calculation shows that Hv,i is ⊕ orthogonal to H when i ≠′ j. Finally it is also clear that H = H . v,j v ∑i<αv v,i Deﬁne a unitary Uv,i ∶ HG,v → Hv,i by setting Uv,iξµ = Sµξv,i and extending linearly. This unitary then implements a unitary equivalence between LG,v ⊥ and SSHv,i . Now deﬁne K = (⊕Hv) , and set T = SSK. Then

(αv) S ≅ T ⊕ ? LG,v . v∈V ∗ From the definition of Wv, it follows that Tv = ∑e∈r 1(v) TeTe for every v ∈ V . Hence T is fully coisometric. Also T is non-degenerate− because S is. ⊕ It is clear from the definition that W = ∑v∈V Wv is a total wandering space. Since the complement of S[W] is fully coisometric, it contains no reducing subspace generated by a wandering space. Hence this space is maximal with this property. Indeed it is unique, because any reducing subspace determined by a total wandering space N has the property that SvN is not in the range of any edge, and thus is contained in Wv. So N ⊂ W. It is clear that the definition of Wv is canonical, and that αv = dim Wv is a unitary invariant. Thus the construction of Hv and K is also canonical. Hence this decomposition is essentially unique up to choice of bases, which is a unitary invariant, and thus two such TCK families S and S′ are unitarily ′ ′ equivalent if and only if αv = αv for v ∈ V and the remainders T and T are unitarily equivalent.

Now we wish to obtain some more precise information about the fully coisometric CK family T in this decomposition. Suppose we have a directed graph G. Let V s ∶= {v ∈ V ∶ r−1(v) = ∅} denote the collection of sources in G. Let G1 be the induced graph on the s vertices V1 = V ∖ V . Proceeding inductively on ordinals, if Gλ is given with vertex set Vλ, we define the graph Gλ+1 to be the graph induced from G on s the vertices Vλ+1 = Vλ ∖ Vλ . When λ is a limit ordinal, we define Gλ to be the graph induced from G on the vertices Vλ = ⋂κ<λ Vκ. Since there are countably many vertices, there is a countable ordinal κG 0 0 such that G ∶= GκG is sourceless. We call G the source elimination of G. Note that since G0 has no sources, a CK family for G0 can be considered as a representation of G which annihilates all vertices in V ∖ V 0. However G0 may have vertices which are infinite receivers. If such a vertex v does ∗ not satisfy Sv = sot–∑r(e)=v SeSe , then there is a wandering space as defined in Theorem 4.2. Once this is eliminated, we obtain a fully coisometric CK family.

Theorem 4.3 (Wold decomposition II). Suppose G = (V,E) is a directed graph, and let G0 = (V 0,E0) be the source elimination of G. Suppose S = (Sv,Se) is a non-degenerate TCK family for G on a Hilbert space H; and STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 19 let (αv) S ≅ T ⊕ ? LG,v v∈V be the Wold-decomposition for S as in Theorem 4.2. Then T is supported 0 0 on V ; and T = (Tv,Te)v∈V 0,e∈E0 is a fully coisometric CK family for G . Proof. By Theorem 4.2 we know that T is a fully coisometric CK family, ∗ so it must satisfy Tv = ∑e∈r 1(v) TeTe for every v ∈ V . So in particular, for s −1 s v ∈ V , we have that Tv = 0.− This means that for any e ∈ s (V ) we have ∗ Te = 0 since 0 = Ts(e) = Te Te. Now, suppose towards contradiction there is some minimal ordinal λ s ′ for which Tw ≠ 0 for some w ∈ Vλ . If λ = λ + 1 is a successor, then s −1 s Tv = 0 for all v ∈ Vλ . Hence, for any e ∈ s (Vλ ) we have Te = 0 since ∗ −1 −1 s 0 = Ts(e) = Te Te.′ Next, since r (w) ⊆ s (Vλ′ ), we must have that ∗ Tw = ∑e∈r 1(w) TeTe = 0, a contradiction. If λ is a′ limit ordinal, we have −1 −1 s −1 s that r (w−) ⊆ s (⋃κ<λ Vκ ). Also for any e ∈ s (⋃κ<λ Vκ ), we have Te = 0. ∗ It follows that Tw = ∑e∈r 1(w) TeTe = 0, again a contradiction. Thus T is supported on V0, and is a− fully coisometric CK family for G0. We can explicitly describe the subspace on which the full CK family lives. Proposition 4.4. Suppose G = (V,E) is a directed graph, and let G0 = 0 0 (V ,E ) be the source elimination of G. Suppose S = (Sv,Se) is a non- degenerate TCK family for G on a Hilbert space H. The space on which the full CK portion of the Wold decomposition (Theorem 4.3) acts is given by: ∗ M = Q SµSµH. k≥1 µ∈F (G),SµS=k Proof. Write the Wold decomposition+ as S = T ⊕ L where T is a full CK family and L is a direct sum of left regular representations. Observe that since T is a full CK family on M, we have that ∗ ∗ IM = Q Tv = Q sot–Q TeTe = sot–Q TeTe . v∈V v∈V r(e)=v e∈E Taking the kth power and using the fact that multiplication on the unit ball is sot continuous, we obtain that k ∗ ∗ ∗ IM = sot–Q TeTe = sot–Q TµTµ ≤ sot–Q SµSµ. e∈E µ∈F (G),SµS=k µ∈F (G),SµS=k It follows that M is contained in the+ intersection in the+ theorem statement. On the other hand, it is completely diﬀerent in the left regular portion. Indeed, in each HG,v ∗ Q SµSµHG,v = Q span{ξν ∶ SνS ≥ k} = {0}. k≥1 µ∈F (G),SµS=k k≥1 µ∈F (G),SµS=k It follows that+ the intersection is exactly+ M. 20 K.R. DAVIDSON, A. DOR-ON, AND B. LI

Deﬁnition 4.5. Let G be a countable directed graph. We say that G has a source elimination scheme (SES) if G0 is the empty graph; i.e., in the elimination procedure described before Theorem 4.3, every vertex appears as a source at some step. Example 4.6. It is clear by construction that a graph with SES must be acyclic. When G is a graph with ﬁnitely many vertices, it is acyclic if and only if it has SES. However the graph G with

V = Z and E = {en ∶ s(en) = n, r(en) = n + 1, n ∈ Z} is an acyclic graph with G0 = G. For directed graphs with SES, it is easy to classify all free semigroupoid algebras up to unitary equivalence. Corollary 4.7. Let G be a directed graph with SES. Then a non-degenerate TCK family S = (Sv,Se) is uniquely determined up to unitary equivalence by the set of multiplicities {αv}v∈V appearing in Theorem 4.3.

Proof. It is clear that αv is a unitary invariant by uniqueness of Wold- decomposition. Since for graphs with SES by Theorem 4.3, if

(αv) T ⊕ ? LG,v v∈V is the Wold decomposition of the TCK family S, then T = (Tv,Te) = (0, 0). Therefore the unitary equivalence class is determined by the multiplicities {αv}v∈V . 5. Single vertex and cycle algebras In this section, we develop two of the main tools that will arise in the proof of the structure theorem in the next section. First, we reduce the existence of wandering vectors to the same problem for free semigroup algebras associated that we associate to every vertex of the graph. Second, we carefully examine the free semigroupoid algebras of a cycle. The following observation will be useful in establishing many properties of free semigroupoid algebras by accessing deep results that have been established for free semigroup algebras. Proposition 5.1. Let S be a free semigroupoid algebra for G = (V,E). Fix + + + v ∈ V , and let F (G)v = {µ ∈ F (G) ∶ s(µ) = r(µ) = v}. Then F (G)v + is the free semigroup generated by the set of irreducible cycles µ ∈ F (G)v.

Moreover, Sv ∶= SvSSSvH is a free semigroup algebra on SvH. + + Proof. Clearly every µ ∈ F (G)v is a product of λ ∈ F (G)v which are + irreducible, and F (G)v is isomorphic to the free semigroup generated by the set of irreducible cycles at v. Next, if µ and ν are two distinct irreducible paths, then Sµ and Sν have ∗ orthogonal ranges. To see this, suppose that SνS ≤ SµS and compute Sν Sµ. This product is zero unless µ = νµ′. However this would mean that µ STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 21 returned to v at an intermediate point—which it does not do. Therefore ∗ Sν Sµ = 0 as claimed. It is evident that wot + Sv = span {SµSSvH ∶ µ ∈ F (G)v} = SvSSSvH is the algebra generated by these irreducible paths. Therefore this is a free semigroup algebra on SvH.

Remark 5.2. Let Iv be the set of irreducible cycles at v. The above proposition is most useful when the number of generators SIvS is at least 2. In this case, results for free semigroup algebras are readily available to us. When Iv = ∅, we have Sv = CSv; and usually this does not cause difficulties. How- ever when SIvS = 1, this means that the transitive component of v is a simple cycle. The analysis here is more complicated. See Theorem 5.6 below for the full story about cycles. The following proposition simplifies the issue of identifying wandering vectors for S, by reducing this problem to free semigroup algebras. Proposition 5.3. Let S be a free semigroupoid algebra generated by a TCK family S = (Sv,Se) for G on H, and let x ∈ H. Then x is a wandering vector for Sv ∶= SvSSSvH if and only if it is a wandering vector for S supported on v. Hence, each SvH is spanned by wandering vectors for Sv if and only if H is spanned by wandering vectors for S. Proof. It is clear that if x is a wandering vector for S supported on v, then it is a wandering vector for Sv. For the converse, we need to verify + that if µ, ν are distinct paths in F (G) with SµS ≥ SνS and s(µ) = s(ν) = v, ∗ ∗ ′ then 0 = ⟨Sµx, Sνx⟩ = ⟨Sν Sµx, x⟩. However Sν Sµ = 0 unless µ = νµ and ′ + ′ µ ∈ F (G)v. Since x is wandering for Sv and Sµ S ≥ 1, we have ⟨Sµ x, x⟩ = 0 as desired. ′ Next, if each SvH is spanned by wandering vectors for Sv, each wandering vector x for some Sv is a wandering vector for S. Therefore H is spanned by wandering vectors for S in ⊕v∈V SvH = H. Conversely, if H is spanned by wandering vectors for S, each such x = Svx is wandering for Sv for every v ∈ supp(x). Hence, each SvH is the span of wandering vectors for Sv. To obtain our structure theorem in the next section, we need to understand free semigroupoid algebras arising from a cycle graph. Let G = Cn be the cycle on n vertices v1, . . . , vn with edges ei with s(ei) = vi and r(ei) = vi+1(mod n) for 1 ≤ i ≤ n. ∞ We let H denote the algebra of all bounded analytic functions on D, ∞ ∞ and let H0 denote the ideal of all functions in H which vanish at 0. We + ∞ introduce the notation Mn (H ) for the set of n×n matrices with coefficients ∞ ∞ in H such that the matrix entries fij ∈ H0 when i > j (i.e., the strictly ∞ lower triangular entries lie in H0 ). This has an operator algebra structure ∞ obtained from considering this as a subalgebra of Mn(L (m)), where m is Lebesgue measure on the circle T. 22 K.R. DAVIDSON, A. DOR-ON, AND B. LI

Lemma 5.4. Consider the left regular representation of the cycle Cn on the n 2 summand HCn,v1 . There is an identiﬁcation of HCn,v1 with C ⊗` such that

Lei ≅ Ei,i+1 ⊗ I for 1 ≤ i < n and Len ≅ En,1 ⊗ U+, where U+ is the unilateral shift. With this identiﬁcation, LCn,v1 is unitarily equivalent to the algebra 2 + ∞ of n × n matrices over B(` ) of the form fij(U+) where fij ∈ Mn (H ).

The algebra LCn is completely isometrically isomorphic and weak-∗ home- + ∞ omorphic to Mn (H ), and the canonical restriction maps to HCn,vi are completely isometric weak-∗ homeomorphic isomorphisms.

Proof. The paths for Cn with s(µ) = v1 are just indexed by their length, say

µk for k ≥ 0. Moreover r(µk) = vk+1(mod n). So the spaces Hvi ∶= Lvi HCn,v1 have orthonormal basis {ξi,s = ξµi 1 ns ∶ s ≥ 0}. Thus there is a natural 2 identiﬁcation of each Hvi with ` .− With+ this basis, we readily identify Lvi with Eii ⊗ I, Lei with Ei,i+1 ⊗ I for 1 ≤ i < n and Len is identiﬁed with En,1 ⊗ U+. A calculation shows that the simple cycle starting and ending at a vertex vi corresponds to Eii ⊗ U+. From this, it follows that for every + + ∞ polynomial matrix pij in Mn (C[z]) ⊆ Mn (H ), there is a corresponding polynomial in the path space that is mapped to pij(U+). Since any element + ∞ [fij] ∈ Mn (H ) is the weak-∗ Cesaro limit of polynomial matrices, [fij] must be mapped to fij(U+) under the unitary equivalence. As the algebra + ∞ of matrices fij(U+) for [fij] ∈ Mn (H ) is clearly a wot-closed subspace 2 of Mn(B(` )), we see that LCn,v1 is unitarily equivalent to the algebra of + ∞ matrices fij(U+) for [fij] ∈ Mn (H ). + ∞ We next show that the algebra of matrices fij(U+) for [fij] ∈ Mn (H ) + ∞ n 2 n 2 is isomorphic to Mn (H ). Since the compression of C ⊗` (Z) to C ⊗` is completely contractive, it is evident that the norms of the matrix operators n 2 n 2 on C ⊗ ` (Z) are at least as large. On the other hand, since C ⊗ ` (Z) is n 2 the closed union of C ⊗` (N0 −k) as k increases, and the restriction to each of these invariant subspaces is unitarily equivalent to LCn,v1 , one sees that the limit algebra is in fact completely isometrically isomorphic and weak-

∗ homeomorphic to LCn,v1 via the canonical map that takes generators to generators. 2 2 Now identify ` (Z) with L (T) by the Fourier transform which implements a unitary equivalence between U to Mz. The von Neumann algebra ∗ ∞ ∞ W (Mz) is the algebra {Mf ∶ f ∈ L (T)} ≃ L (T), where this is a normal ∗-isomorphism. Then LCn,vj is unitarily equivalent to the subalgebra of ∞ + ∞ Mn(L (T)) corresponding to Mn (H ). So this map is completely isometric and a weak-∗ homeomorphism. Hence, this yields a completely isometric and weak-∗ homeomorphic isomorphism between LCn,v1 and LCn,v1 . Finally, since for each 1 ≤ i ≤ n we similarly have a completely isometric + ∞ and weak-∗ homeomorphic isomorphism ϕi ∶ Mn (H ) → LCn,vi , the rest follows from Corollaries 3.2 and 3.3. A similar analysis shows that a general CK family depend only on a single unitary. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 23

Lemma 5.5. Let S = (Sv,Se) be a CK family for Cn. Then there is an n identiﬁcation of H with C ⊗ K and a unitary operator V on K such that

Sei ≅ Ei,i+1 ⊗ I for 1 ≤ i < n and Sen ≅ En,1 ⊗ V . Proof. This follows in a similar manner to the previous lemma. We denote

Hvi = Svi H, and use Sei to identify Hvi with Hvi 1 for 1 ≤ i < n. The H H only diﬀerence now is that the unitary edge Sen mapping+ vn onto v1 will be identiﬁed with an arbitrary unitary operator V instead of the unilateral shift. We can split the spectral measure of V into a portion which is absolutely continuous with respect to Lebesgue measure m and one that is singular, say V ≅ Va ⊕ Vs on K = Ka ⊕ Ks. This yields a corresponding decomposition n n of the CK family S on (C ⊗ Ka) ⊕ (C ⊗ Ks) into S ≅ Sa ⊕ Ss where Sa is its absolutely continuous part and Ss is its singular part. This allows us to describe general free semigroupoid algebras for a cycle.

Theorem 5.6. Let S = (Sv,Se) be a TCK family for Cn. Write its Wold decomposition as S = T ⊕ ∑n ⊕L(αi) where T is CK. Let V be the unitary i=1 Cn,vi from Lemma 5.5 in the structure of T. Decompose V = Va ⊕ Vs where Va is a unitary with spectral measure µa absolutely continuous with respect to Lebesgue measure m, and Vs is a unitary with spectral measure µs singular with respect to Lebesgue measure m. Then there are two situations,

(1) If S is a CK family (i.e. αi = 0 for 1 ≤ i ≤ n) and m ≪~ µ, then S is ∞ isomorphic to the von Neumann algebra Mn(L (µ)). n ∞ (2) If ∑i=1 αi ≥ 1 or m ≪ µ, then S is isomorphic to LCn ⊕Mn(L (µs)). All of these isomorphisms are completely isometric weak-∗ homeomorphic isomorphisms. Furthermore, in case (1) there are no wandering vectors, and in case (2) the wandering vectors span the left regular part and the absolutely continuous portion of the CK part. Proof. Suppose S is a CK family with unitary V as above with spectral measure µ = µa + µs. A theorem of Wermer [50] shows that the wot-closed algebra W (V ) is isomorphic to L∞(µ) if m ≪~ µ, and W (V ) is isomorphic ∞ ∞ ∞ to H ⊕L (µs) if m ≪ µ. So when m ≪~ µ, it follows that S ≃ Mn(L (µ)). ∞ ∗ When m ≪ µ, the wot-closure of Alg(V ) is H (Va) ⊕ W (Vs), where V ≅ Va ⊕ Vs is the decomposition of V into its absolutely continuos and singular parts. Since µa is equivalent to m, it follows that the bilateral shift U is unitarily equivalent to a direct summand of Va and that Va is unitarily equivalent to a direct summand of U (∞). By Lemma 5.4 we see ∞ that S ≃ LCn ⊕ Mn(L (µs)). Next, consider the case in which there is a left regular part and a CK part. Wermer’s analysis [50] of the wot-closed algebra of an isometry shows that ∞ ∞ the wot-closed algebra W (U+ ⊕V ) is isomorphic to H ⊕L (µs). By using ∞ Lemma 5.4 again, we see that S ≃ LCn ⊕ Mn(L (µs)). 24 K.R. DAVIDSON, A. DOR-ON, AND B. LI

In case (1), the algebra is self-adjoint and thus cannot have wandering vectors. Indeed, if there was a wandering vector, it would yield an invariant subspace on which the algebra S is unitarily equivalent to a nonself-adjoint algebra. This will contradict the previous analysis. In case (2), the argument from the previous paragraph shows that the wandering vectors are contained in the left regular and absolutely continuous n n parts of the space, which we identified with C ⊗Kl and C ⊗Ka respectively. In order to show that this part of the space is the closed span of wandering vectors, we turn to Proposition 5.3, which reduces the problem to showing for each 1 ≤ i ≤ n that the free semigroup algebra Svi has Kl ⊕ Ka as the closed span of wandering vectors. Clearly Kl is spanned by wandering vectors. If µa ≈ m, then by the proof of [12, Theorem II.1.3], and perhaps after splitting and unifying subsets, we can arrange for ⊕ Va ≅ Q Mz,Ai where T = A0 ⊇ Ai ∀i > 0, i≥0 2 where Mz,A is multiplication by z on L (A). Then for i > 0 2 2 2 c 2 2 L (A0) ⊕ L (Ai) ⊇ L (Ai ) ⊕ L (Ai) ≅ L (T). Thus, this is a reducing subspace on which the restriction of V is unitarily equivalent to Mz, and is thus is spanned by wandering vectors. Since the 2 c 2 union of subspaces {L (Ai ) ⊕ L (Ai)}i≥0 has dense span in Ka (up to the above unitary identification), we see that Ka is also spanned by wandering vectors. In the remaining case, some αi ≥ 1 and µa ≈ mSA for some measureable subset A ⊆ T. Thus Svi is generated by an isometry V which, modulo 2 2 2 multiplicity, has the form V = U+ ⊕ Mz,A ⊕ Vs on H = H ⊕ L (A) ⊕ L (µs). In this case, the wandering vectors span a dense subset of H2 ⊕L2(A). This is a well-known fact, but we review the argument. Let ε > 0 and fix k ∈ Z. By strong logmodularity of H∞, there is a function h ∈ H∞ ⊆ H2 such that 2 1~2 k ShS = χAc + εχA. Let x = h ⊕ (1 − ε) z χA ⊕ 0. Then if m = l + p, and p > 0, ⟨Smx, Slx⟩ = ⟨Spx, x⟩ p 1~2 k+p 1~2 k = az h ⊕ (1 − ε) z χA ⊕ 0, h ⊕ (1 − ε) z χA ⊕ 0f p 2 p p = S z ShS + (1 − ε) S z χA = S z = 0. It follows that the wandering vectors span H2 ⊕ L2(A).

Example 5.7. Consider two CK families on Cn as in Lemma 5.5 with unitaries Mz,A and Mz,Ac , where A is a subset of T with 0 < m(A) < 1 and c A = T ∖ A. Then by Theorem 5.6(1), they both generate free semigroupoid algebras which are von Neumann algebras. However their direct sum forms the CK family with unitary Mz, which has spectral measure m. This yields a representation with wandering vectors and is hence nonself-adjoint. This shows that when the graphs are cycles, there are nonself-adjoint free semigroupoid algebra with complementary reducing subspaces so that STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 25 the cutdown by each is a von Neumann algebra with absolutely continuous associated isometries Mz,A and Mz,Ac (see Section7).

6. The structure theorem In this section, we establish a structure theorem for free semigroupoid algebras of a graph G akin to the one in [14], along with a complete iden- tiﬁcation of wandering vectors. Proposition 6.1. Let G be a countable directed graph, and let S be the free semigroupoid algebra generated by a non-degenerate TCK family S = (Sv,Se). Suppose that ϕ ∶ S → LG is a wot-continuous and completely + contractive homomorphism such that ϕ(Sλ) = Lλ for all paths λ ∈ F (G). Then ϕ is a surjective complete quotient map. Hence, S~ ker ϕ is completely isometrically isomorphic and weak-∗-homeomorphic to LG. Proof. By the universality of T (G), we have a surjective ∗-homomorphism + πS ∶ T (G) → TS given by πS(Lλ) = Sλ for all λ ∈ F (G). Hence, if X ∶=

∑` A` ⊗ Sλ` is a ﬁnite sum in AS, then YXY ≤ YAY where A = ∑` A` ⊗ Lλ` . For A ∈ Mp(LG) with YAY ≤ 1, consider its Fourier series expansion SλS ∑ Aλ ⊗ Lλ, and the Ces`aromeans Σk(A) = ∑SλS

YXk,F Y ≤ YΣk(A)PF Y ≤ 1. (p) Moreover, ϕ (Xk,F ) = Σk(A)PF . Since the ball of radius YAY in S is wot-compact, there is a coﬁnal subnet Xα of {Xk,F } that converges wot to some element X ∈ S. By the wot- continuity of ϕ(p), we see that ϕ(p)(X) = A. By complete contractivity of ϕ, we have YXY ≤ YAY = Yϕ(p)(X)Y ≤ YXY. Thus the map ϕ is surjective, and the map

ϕ˜ ∶ S~ ker ϕ → SG is a completely isometric isomorphism. Since ϕ is wot-continuous and both S and ker ϕ are weak-∗ closed, we see that ϕ̃ is weak-∗ to wot continuous. However, the weak-∗ and wot topologies on LG coincide by [32, Corollary 3.2], and so ϕ̃ is weak-∗ to weak-∗ continuous. Using the fact that ϕ is an isometry, the usual Banach space pre-duality arguments show that the inverse of ϕ̃ is also weak-∗ to weak-∗ continuous. Therefore ϕ̃ is a weak-∗ homeomorphism. 26 K.R. DAVIDSON, A. DOR-ON, AND B. LI

We recall that for a TCK family S = (Sv,Se), the support of S is given by supp(S) = {v ∈ V S Sv ≠ 0}, which is a directed subset of V ; and we denote by G[S] ∶= G[supp(S)] the subgraph induced from G on the support of S. Definition 6.2. We say that a free semigroupoid algebra S generated by a TCK family S = (Sv,Se) is analytic type if there is a completely isometric isomorphism and weak-∗-homeomorphism ϕ ∶ S → LG[S] such that ϕ(Sλ) = + Lλ for any λ ∈ F (G[S]). By Proposition 6.1 it suffices to require that ϕ above only be a wot- continuous completely contractive injective homomorphism. Moreover, if G is a transitive graph, then the only non-empty directed subgraph of G is G itself. So if S is of analytic type, then it must be isomorphic to LG. For a wandering subspace W ⊆ H of a free semigroupoid algebra S, we will denote by G[W] the graph G[supp(W)]. For a wandering vector x ∈ H, we will also denote G[x] for the graph G[{x}]. To streamline the presentation of our results, we make the following definition. Definition 6.3. Let G be a directed graph and S the free semigroupoid algebra of a TCK family S = (Sv,Se) on a Hilbert space H. We say that a vertex is wandering for S if there is a non-zero wandering vector for S supported on v. We will denote by Vw the collection of all wandering vertices for S. When x ∈ H is a wandering vector supported on v for a free semigroupoid + algebra S, and Sλx ≠ 0 for a path λ ∈ F (G) with s(λ) = v, then Sλx is a wandering vector supported on r(λ). Hence, it is easy to see that Vw is directed. Corollary 6.4. Let S be a free semigroupoid algebra of a graph G generated by a TCK family S = (Sv,Se) on H. If S has a wandering subspace W ⊆ H, then there is a wot-continuous completely contractive homomorphism ϕ ∶ S → LG[W] obtained via the restriction map to the subspace S[W] which is surjective, and is a complete quotient map. Hence, the induced map ϕ̃ ∶ S~ ker ϕ → LG[W] is a completely isometric weak-∗ homeomorphic isomorphism. Moreover, when M is the closed span of all wandering vectors for S, then SSM is completely isometrically isomorphic and weak-∗-homeomorphic to LG[Vw]. In particular, if H is the closed span of wandering vectors, then S is analytic type. ⊕ Proof. Since W is wandering, S[W] = ∑v∈supp(W) S[SvW]. Let M = S[W], and for v ∈ supp(W), fix a unit vector xv in SvW. Let Uv ∶ HG,v → M be given by Uvξλ = Sλxv for λ ∈ F+(G) with s(λ) = v, which is an isome- ∶ ⊕ try. Then U = ∑v∈supp(W) Uv is an isometry of HG,supp(W) into H. The map ϕ(A) = U ∗AU is a wot-continuous completely contractive homomor- S phism that takes Sλ to Lλ HG,supp . By Proposition 6.1, we get that

(W) STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 27

ϕ ∶ S → LG[W] and the induced map ϕ̃ have the properties required in the statement. ∶ Next, we construct a map ψ SSM → LG[Vw] given by ψ(SλSM) = Lλ for + λ ∈ F (G) that will be completely contractive and wot continuous. Indeed, if {xi}i∈I is the collection of all wandering vectors, we have a completely con- ∶ S → tractive and wot continuous maps ϕi S M LG[supp(xi)]. After restricting, Corollary 3.2 yields that for each v ∈ Vw ∶= ⋃i∈I supp(xi) we have a well- defined completely contractive and wot continuous map ϕv ∶ SSM → LG,v + that maps SλSM to Lλ for each λ ∈ F (G). Since each HG,v is reducing for ∶ LG[Vw], we see that ψ = ⊕v∈Vw ϕv SSM → LG[Vw] is a well-defined completely contractive wot continuous map. Thus, by Proposition 6.1 it will suffice to show that ψ is injective. For any T ∈ SSM and a wandering vector x ∈ H, let Tx ∶= T SS[x] be the restriction. ( ) = ( )S = If ψ T 0, it is clear that ψ T HG,supp x 0. So by the first paragraph we ( )S = know that Tx is unitarily equivalent to( ψ) T HG,supp x ; whence Tx 0. As ∈ M this was done for an arbitrary wandering vector x ( ) , we conclude that T SM = 0. Therefore, the map ψ is an injective wot-continuous completely contractive homomorphism. Thus by Proposition 6.1 it is a completely isometric isomorphism that is also a weak-∗-homeomorphism. Definition 6.5. For a free semigroupoid algebra S of a graph G generated by a TCK family S = (Sv,Se), we define wot wot S0 ∶= ⟨Se⟩ = span {Sλ ∶ SλS ≥ 1} to be the wot-closed ideal generated by the edge partial isometries. Simi- larly we define k wot wot S0 ∶= ⟨Sµ ∶ SµS = k⟩ = span {Sλ ∶ SλS ≥ k}. k In particular, we have the ideals LG,0 in LG. Lemma 6.6. For k ≥ 1, k S0 = sot–Q SλAλ ∶ Aλ ∈ S. SλS=k k Each element of S0 has a unique such decomposition when Aλ = Ps(λ)Aλ.

Proof. The algebraic ideal Ik generated by {Sλ ∶ SλS = k} consists of polynomials ∑SµS≥k aµSµ, where aµ = 0 except finitely often. By combining terms ′ for µ of the form µ = λµ for SλS = k, we get an expression ∑SλS=k SλAλ where ∗ Aλ is a polynomial in T+(G). Moreover, one computes that Aλ = SλA. k Given A ∈ S0, there is a net Aα ∈ Ik wot-converging to A. We write Aα = ∑SλS=k SλAα,λ. Then define ∗ ∗ Aλ ∶= SλA = wot–lim SλAα = wot–lim Aα,λ. ∗ Thus Aλ lie in S. The projection P = sot–∑SλS=k SλSλ satisfies PSµ = Sµ for + all µ ∈ F (G) with SµS ≥ k. It follows that PAα = Aα for all α, and therefore 28 K.R. DAVIDSON, A. DOR-ON, AND B. LI

PA = A. Hence ∗ A = PA = sot–Q SλSλA = sot–Q SλAλ. SλS=k SλS=k ∗ The Aλ are uniquely determined by the equation Aλ = Ps(λ)Aλ = SλA. ∗ k k−n Corollary 6.7. If SµS = n ≤ k, then SµS0 ⊆ S0 , including k = n where we 0 interpret S0 to be S. Proof. Note that if SλS = k, then ⎪⎧ ′ ∗ ⎪Sλ if λ = µλ SµSλ = ⎨ ⎪ ′ ⎩⎪0 otherwise. k Therefore if A in S0 is written A = sot–∑SλS=k SλAλ, we obtain that ∗ SµA = sot–Q Sλ Aµλ Sλ S=k−n ′ ′ k−n ′ which lies in S0 . Theorem 6.8. Let S be the free semigroupoid algebra of a TCK family S = (Sv,Se) on a Hilbert space H for a graph G. Let V0 = {Sv ∶ Sv ∈ S0}. Then Vw = V ∖ V0, and we have a completely isometric isomorphism and weak-∗ homeomorphism ∞ S~S0 ≅ ` (Vw). = Moreover, if we denote by P0 sot–∑v∈V0 Sv, then we have that ⊥ (1) P0 H is invariant for S. (2) P0SP0 is self-adjoint.

Proof. First, we verify that Vw = V ∖ V0. If there’s a wandering vector supported at v, by Corollary 6.4 there is a completely contractive wot continuous map from S to LG,v, so that clearly v ∈ V ∖ V0.

Conversely, for each v ∈ V , by Proposition 5.1 the algebra Sv ∶= SvSSSvH is a free semigroup algebra with generators indexed by the irreducible cycles around v. By Proposition 5.3 there is a wandering vector for S supported on v if and only if the free semigroup algebra Sv ∶= SvSSSvH has a wandering vector. If v ∈ V ∖ V0 and there are no cycles at v, then any vector in SvH is wandering for Sv, and hence is wandering for S and supported at v by Proposition 5.3. If v ∈ V ∖ V0 and there are at least 2 irreducible cycles at v, then Sv is a free semigroup algebra on 2 or more generators. Since Sv ≠ Sv,0, Sv has a wandering vector by Kennedy’s work [35, Theorem 4.12]; and again this provides a wandering vector for S supported at v. Finally, if v ∈ V ∖ V0 and there is exactly one irreducible cycle at v, then the compression to this cycle is a free semigroupoid algebra T classiﬁed by Theorem 5.6. Since T ≠ T0, we see that T is not a von Neumann algebra, and hence lies in cases (2) of Theorem 5.6, which yields the desired wandering STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 29 vector. Therefore we can ﬁnd a wandering vector for S for every v ∈ V ∖ V0, and hence Vw = V ∖ V0. Now consider (1). Clearly, SvP0H ⊆ P0H for all v ∈ V . Next, if e ∈ E is ∗ such that r(e) ∈ V0, then Se P0H = {0} ⊆ P0H. So assume r(e) ∈ V0. As Vw is directed, we must have that s(e) ∈ V0, so that ∗ ∗ Se P0H = Ps(e)Se P0H ⊆ Ps(e)H ⊆ P0H. ∗ ⊥ Thus P0H is invariant for S ; whence P0 H is invariant for S. For item (2), let e ∈ E be such that P0SeP0 = Se. In particular, this means that r(e) ∈ V0, so that Sr(e) ∈ S0. Hence ∗ ∗ ∗ Se = Ss(e)Se Sr(e) ∈ P0(Se S0)P0 ⊆ P0SP0 by Corollary 6.7. Therefore P0SP0 is self-adjoint. Finally, observe that

wot ∞ wot S = span {Sλ ∶ SλS ≥ 0} = ` (V ∖ V0) + S0 . Deﬁne a map ⊥ Ψ(A) = P0 Φ0(A) = wot–Q avSv v∈V ∖V0 where Φ0 is the 0-th order Fourier coeﬃcient map. This is clearly a com- ∞ pletely contractive weak-∗ continuous map onto ` (V ∖ V0). The kernel contains span{Sλ ∶ SλS ≥ 1} and thus contains S0. Also for any polynomial A, it is clear that A−Ψ(A) belongs to S0. Taking limits shows that A−Ψ(A) ∞ belongs to S0 in general. Therefore S = ` (V ∖ V0) + S0 and ker Ψ = S0. ∞ The natural injection of ` (V ∖ V0) into S is a completely isometric section of this map. It follows that Ψ is a complete quotient map, and induces a ∞ completely isometric weak-∗ homeomorphism of S~S0 and ` (Vw).

Remark 6.9. To get that Vw = V ∖ V0 in the above proof we apply a deep result of Kennedy [35]. We note that results in [35, 36] are true for free semigroup algebras with ℵ0 generators. It may appear on a casual reading that d ≥ 2 must be ﬁnite, but a careful look at the proofs and the cited references shows that they work for d = ℵ0 as well. The above result can be considered as a generalization of the Dichotomy in [14, Theorem 1.5]. Actual dichotomy is obtained when the graph G is transitive. Corollary 6.10 (Dichotomy). Let S be a free semigroupoid algebra of a transitive graph G = (V,E) generated by a TCK family S = (Sv,Se) on H. Then either S is a von Neumann algebra or there is a wot-continuous completely contractive homomorphism ϕ ∶ S → LG such that ϕ(Sλ) = Lλ for + every λ ∈ F (G). Proof. Since G is transitive, the only directed subgraphs are G and the empty graph, so that either Vw = ∅ or Vw = V . In the ﬁrst case S = P0SP0 is a von Neumann algebra, and in the second case there is a wot-continuous 30 K.R. DAVIDSON, A. DOR-ON, AND B. LI completely contractive homomorphism ϕ ∶ S → LG such that ϕ(Sλ) = Lλ for + every λ ∈ F (G). Lemma 6.11. Suppose that G is a graph and S is nonself-adjoint. Denote [ ] ∶= ′ = ( ) ′ ∶= ⊥ ⊥ ∶ → G Vw G Vw,Ew , S P0 SP0 , and let ϕ S LG[Vw] be the canonical surjection given in Corollary 6.4. Then every element A ∈ S′ can be uniquely represented as

A = Q akSλ + Q SµAµ SλS

A = Q aλSλ + Q SµAµ SλS

∗ So J is an ideal of S which is invariant under left multiplication by any Sµ. Thus J is a left ideal of M. By basic von Neumann algebra theory, there is a projection P ∈ M such that J = MP . However this lies in S, so P ∈ S. Hence S contains the self-adjoint algebra P MP . For v ∈ V , since Sv ∈ S, J = MP contains JSv = MPSv. It follows that PSv = PSvP , and since the right term is self-adjoint, SvP = PSv. This corollary characterizes when S is self-adjoint. There is one excep- tional case, the trivial graph consisting of a vertex with no edges. Corollary 6.13. Let S be the free semigroupoid algebra on H generated by a TCK family S = (Sv,Se) for G. Assume that the graph GS = G[supp S] has no trivial components. Then the following are equivalent:

(1) S0 = S. (2) S is a von Neumann algebra. (3) The set Vw is empty. In this case, GS must be a disjoint union of transitive components.

Proof. We may assume that G = GS. k The hypothesis S0 = S implies that S0 = S for every k ≥ 1. Therefore J = S contains I, so that P = I and S = M is self-adjoint. Assume that S is self-adjoint. If the connected components of G are not transitive, then there is an edge e ∈ ES such that v ∶= s(e) and w = r(e) such that w ≻~ v. But then 0 ≠ Se = PWw SePWv ∈ S and PWv SPWw = {0}. This implies that S is not self-adjoint, a contradiction. Hence G must be the disjoint union of its transitive components. Since G has no trivial components and each component is transitive, for each vertex v ∈ V , there is some edge e with s(e) = v. As S is self-adjoint, ∗ there is a net Aλ of polynomials converging in the wot topology to Se . ∗ Consequently Sv = Se Se = wot–lim AλSe belongs to S0. Hence, Vw = ∅. Finally if Vw is empty, Theorem 6.8 shows that S~S0 ≅ {0}; so that S0 = S. Remark 6.14. The case of a trivial graph G is pathological because every non-degenerate free semigroupoid algebra on G consists of scalars. It is self-adjoint, but S0 = {0} ≠ S. If S is a TCK family such that GS = G[supp S] is not all of G, then the hypothesis of no trivial components applies to GS. For example, if V = {v, w} and E = {e} with s(e) = v and r(e) = w, consider the TCK family Sv = Se = 0 and Sw = 1 on C. Then S = C is self-adjoint and S0 = {0} because GS is the trivial graph. We can now provide a structure theorem for free semigroupoid algebras which is analogous to the structure theorem for free semigroup algebras [14]. Theorem 6.15 (Structure theorem). Let S be the free semigroupoid algebra ∗ on H generated by a TCK family S = (Sv,Se) for G. Let M = W (S) be the ∶ von Neumann algebra generated by S. When Vw ≠ ∅, let ϕ S → LG[Vw] be 32 K.R. DAVIDSON, A. DOR-ON, AND B. LI the homomorphism of Corollary 6.4. Then there is a projection P ∈ S∩{Sv ∶ ∈ }′ = v V that dominates the projection P0 ∑v∈V0 Sv such that k (1) MP = ⋂k≥1 S0 = J. (2) If Vw ≠ ∅ then ker ϕ = J. (3) P ⊥H is invariant for S. (4) With respect to H = P H ⊕ P ⊥H, there is a 2 × 2 block decomposition P MP 0 S = P ⊥MP SP ⊥ ⊥ (5) If Vw ≠ ∅, then SP is completely isometrically and weak-∗-homeo-

morphically isomorphic to LG[Vw]. (6) P ⊥H is the closed linear span of the wandering vectors. (7) If there’s a cycle through every every vertex v ∈ V , then P is the largest projection such that P SP is self-adjoint. Proof. We may assume on the outset that G is equal to the induced subgraph GS on supp(S), so that Sv ≠ 0 for every v ∈ V . Next, let us dispose of the case of trivial components. If G is the trivial graph, with V = {v} and E = ∅, then a TCK family for G consists only of the projection Sv. Indeed, if S is non-degenerate, then Sv = I. Thus S = CSv, and the ideal S0 = {0}. The projection P = 0 and items (2–6) are trivial, and (7) does not apply. For the rest of the proof, up to decomposing G as a disjoint union of undirected connected components, we assume there are no trivial components. If S is self-adjoint (and G has no trivial components), by Corollary 6.13 k this is equivalent to S0 = S for all k ≥ 1. Hence P = I, and items (1) through (4), (6) and (7) are readily veriﬁed. Hence, we may assume for the rest of the proof that S is nonself-adjoint, and that Vw is non-empty. k By Lemma 6.12, we get that MP = ∩k≥1S0 and P commutes with each k k Sv. Hence, for every Sv ∈ S0 we have Sv = Sv ∈ ⋂k≥1 S0 = J, so that P ≥ Sv for v ∈ V0 and hence P ≥ P0. Lemma 6.11 shows that J = ϕ−1(Lk ) = ϕ−1 Lk = ker ϕ. G[Vw],0 G[Vw],0 k≥1 k≥1 Since ϕ is non-zero, J ≠ S and so P ≠ I. As for (3), observe that P H = P MH = J∗H. Thus ∗ ∗ ∗ ∗ SλP H = SλJH = (JSλ) H ⊆ J H = P H Therefore P H is invariant for S∗, whence P ⊥H is invariant for S. This shows that the 1, 2 entry of the matrix in (4) is 0. Now SP ⊆ MP = J = JP ⊆ SP. Hence SP = MP = P MP + P ⊥MP form the 1, 1 and 2, 1 entries of the matrix. That leaves the 2, 2 entry P ⊥SP ⊥ = SP ⊥ by the invariance of P ⊥H. For item (5), Proposition 6.1 shows that ϕ induces a completely isometric ∶ weak-∗-homeomorphic isomorphism ϕ̃ S~J → LG[Vw]. However, as J = STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 33

SP , we see that S~J ≅ SP ⊥. Therefore SP ⊥ is completely isometrically isomorphic and weak-∗ homeomorphic to LG[Vw]. For item (6), by Proposition 6.4 the closed linear span of the wandering ⊥ vectors is contained in P H. For the converse, as P commutes with Sv for every v ∈ V , by Proposition 5.1 and Proposition 5.3 it will suffice to show ⊥ that the closed linear span of non-zero wandering vectors for Sv is P SvH for each v ∈ V . There are three cases to be handled. The first case is when there are no cycles at the vertex v, so that Sv ≅ C. In this case, every vector in ⊥ P SvH is wandering. The second case is when Sv is an analytic type free semigroup algebra generated by at least two irreducible cycles. In this case [16, Corollary 2.5] and [16, Lemma 3.2] reduce the problem to establishing ⊥ the existence of a wandering vector in P SvH (where again the proofs work for ℵ0 generators). By Theorem 6.8, Vw = V , which provides such a wandering vector at each vertex. The third case occurs when there is a single irreducible cycle at v. Here we apply Theorem 5.6 to conclude that the ⊥ wandering vectors span the complement of the von Neumann part P Hv. For item (7), assume every vertex has a cycle through it and suppose that Q ∈ S is a projection such that QSQ is self-adjoint. Decompose H = P H ⊕ P ⊥H to obtain a 2 × 2 decomposition for Q as in (4). Since Q is self- adjoint, its 2, 1 corner must vanish, and hence P commutes with Q. So by ⊥ (5), P Q can be identified as a self-adjoint element of LG[Vw] via ϕ. By [37, ⊥ Corollary 4.6], ϕ(P Q) must be the sum of projections ∑v∈F Lv for some ⊥ ⊥ subset F ⊆ Vw. Hence P Q = ∑v∈F SvP . If F were non-empty, there would ∗ be a cycle µ with s(µ) = r(µ) = v ∈ F . Now Sµ does not belong to S, for if ∗ it did, then Sv = SµSµ would lie in S0 and so v ∈ V0 = V ∖ Vw, contrary to ∗ ∗ ⊥ hypothesis. As SµP does belong to S, we see that SµP does not belong to QSQ. Thus QSQ is not self-adjoint. It follows that F must be empty and Q ≤ P . In other words, P is the largest projection such that P SP is self-adjoint. Remark 6.16. In [41, Theorem 7.8] Muhly and Solel provide a structure theorem that works in the general context of W ∗-correspondences. Their theorem applies to our context when the graph G has finitely many vertices and edges to obtain items (1) - (5) in our theorem above. Our proof avoids W ∗-correspondence machinery, and yields items (6) and (7).

7. Absolute continuity and wandering vectors In [16], a notion of absolute continuity for representations of the Toeplitz- Cuntz algebra was introduced and used to tackle the question of the existence of wandering vectors for analytic type representations. Absolute continuity was then further investigated by Muhly and Solel [41] in the context of W ∗-correspondences, where a representation theoretic characterization was found. This characterization was then used by Kennedy in [36] where he showed that every absolutely continuous representation of a free semigroup 34 K.R. DAVIDSON, A. DOR-ON, AND B. LI algebra on at least two generators is analytic type. In this section we obtain analogues of these results for free semigroupoid algebras.

Deﬁnition 7.1. A linear functional ϕ on T+(G) is absolutely continuous if there are vectors ξ, η ∈ HG so that ϕ(A) = ⟨πL(A)ξ, η⟩. A TCK family S (or the free semigroupoid algebra S that it generates) associated to a representation π of T (G) on H is absolutely continuous if the functionals ϕ(A) = ⟨πS(A)x, y⟩ on T+(G) are absolutely continuous for every x, y ∈ Hπ. Also a vector x ∈ H is absolutely continuous if ϕ(A) = ⟨π(A)x, x⟩ is absolutely continuous.

We note that the restriction to S[W] for a wandering subspace W is always absolutely continuous, as that space is unitarily equivalent to a left regular type. Another important observation is that ϕ ∶ T+(G) → C is absolutely continuous if and only if it extends to a weak-∗ continuous on LG. Indeed, if it is absolutely continuous, then it clearly extends. Conversely, when ϕ extends, as LG has property A1 by [32, Theorem 3.1], we see that ϕ is a vector functional. We note that the assumption of no sinks in [32, Theorem 3.1] is unecessary as the Wold decomposition of Theorem 4.2 can be used instead in its proof. In particular, we see that the set of absolutely continuous functionals is identiﬁed with the predual of LG. Hence it is complete, and thus forms a norm-closed subspace of the dual of T+(G). Also, from [32, Corollary 3.2], we see that the weak and weak-∗ topologies coincide on LG. The following explains why absolute continuity for free semigroupoid algebras is natural analogue of the operator theory fact that any absolutely continuous isometry has an H∞ functional calculus.

Proposition 7.2. Let G be a directed graph and S a free semigroupoid algebra generated by a TCK family S. Let πS ∶ T (G) → B(H) be the representation associated to S. Then the following are equivalent (1) S is absolutely continuous. (2) The representation πS ∶ T+(G) → B(H) extends to a completely contractive weak-∗ continuous representation of LG.

Proof. Suppose (2) holds and that ϕ is given by ϕ(A) = ⟨πS(A)x, y⟩ for x, y ∈ H and A ∈ T+(G). By hypothesis, ϕ extends to a weak-∗ continuous functional on LG. Hence ϕ is absolutely continuous. For the converse, suppose T ∈ LG. By Remark 2.1 we can find a net Xα in T+(G) in the YT Y-ball such that πL(Xα) converges to T in the weak operator topology on LG. In order to define the extension of πS to LG, it will suffice to show that πS(Xα) converges wot to some element Y ∈ B(H). Since πS(Xα) is a bounded net, it has a wot cluster point Y . If x, y ∈ H, let ξ, η ∈ HG be chosen so that

ϕ(A) ∶= ⟨πS(A)x, y⟩ = ⟨πL(A)ξ, η⟩. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 35

Then Y satisﬁes

⟨Y x, y⟩ = lim⟨πS(Xα)x, y⟩ = lim⟨πL(Xα)ξ, η⟩ = ⟨T ξ, η⟩.

Therefore Y is uniquely determined, and thus wot–limα πS(Xα) = Y . So we may deﬁne the extension πS ∶ LG → B(H) by

πS(T ) = wot–lim πS(Xα). α A similar argument shows that this is independent of the choice of the net Xα. This shows that this extension is wot-continuous. Note that since πS(Xα) is in the YT Y-ball of B(H), we get that πS is contractive on LG.A similar argument for matrices shows that πS is completely contractive. Following the approach for free semigroup algebras in [16], we also deﬁne intertwining maps.

Deﬁnition 7.3. Let S = (Sv,Se) be a TCK family for G on H. An inter- + twiner is a map X ∈ B(HG, H) satisfying SµX = XLµ for µ ∈ F (G). We let X (S) denote the set of all intertwining maps. Let

Vac(S) = {x = Xη ∶ X ∈ X (S)} be the union of the ranges of intertwining maps. The following is a straightforward generalization of [14, Theorem 1.6].

Lemma 7.4. Suppose that S = (Sv,Se) is a TCK family for a graph G on H, and let x ∈ H be an absolutely continuous unit vector. Let T = S ⊕ L = (Sv ⊕ Lv,Se ⊕ Le) generate the free semigroupoid algebra T on H ⊕ HG. For any ε > 0, there exists a vector ζ ∈ HG with YζY < ε such that the restriction of T to the cyclic subspace N ∶= T[x ⊕ ζ] is unitarily equivalent to LG,V for ′ some subset V ⊆ V . ′

Proof. Let πS be the representation of T (G) associated to S. Consider the + + vector state on T (G) given by ψ(A) = ⟨πS(A)x, x⟩ for A ∈ T (G). Since x is absolutely continuous, there are vectors ζ and η in HG such that

ψ(A) = ⟨πL(A)ζ, η⟩. As ψ(A) = ⟨ϕ(S) tζ, t−1η⟩ for any non-zero scalar t, we may assume that YζY < ε. We then compute for all A ∈ T +(G),

⟨(πS(A) ⊕ πL(A))(x ⊕ ζ), x ⊕ −η⟩ = ⟨πS(A)x, x⟩ − ⟨πL(A)ζ, η⟩ = 0.

So x⊕−η is orthogonal to the cyclic subspace N ∶= T[x⊕ζ]. Let N = (Nv,Ne) be the restriction of T to N , and let N be its free semigroupoid algebra. By the Wold decomposition for N, we obtain an orthogonal decomposition N = L + M where L is the left regular part determined by the wandering ∗ subspaces Wv = (Nv − ∑e∈r 1(v) NeNe )N , so that − L ∶= Q T[Wv] v∈V 36 K.R. DAVIDSON, A. DOR-ON, AND B. LI and M is the fully coisometric CK part, given by Proposition 4.4 as k M ∶= T0[x ⊕ ζ]. k≥1 k Now, since T0(H ⊕ HG) ⊆ Hs ⊕ QkHG where Qk is the projection onto the span of {ξµ ∶ SµS ≥ k}. These spaces have intersection H ⊕ 0. Thus M is contained in H ⊕ 0. Since x ⊕ ζ belongs to N = M + L, we may write x ⊕ ζ = y + z, where y = y0 ⊕ 0 ∈ M and z = (x − y0) ⊕ ζ lies in L. Moreover, y and z are orthogonal, whence 0 = ⟨y0, x − y0⟩. We saw above that x ⊕ −η is orthogonal to N = M + L, and, in particular, is orthogonal to y0 ⊕ 0. Therefore we also have ⟨x, y0⟩ = 0. Hence ⟨y0, y0⟩ = 0, i.e., y0 = 0. Therefore, x ⊕ ζ is contained in L. But L is invariant for T, and x ⊕ ζ generates N ; whence N = L and M = 0. Thus, since N is a cyclic subspace for T generated by a single vector, the dimension of Wv is at most one for ′ each v ∈ V . Deﬁne V to be those v ∈ V for which Wv is of dimension ⊕ 1. Therefore, N is unitarily equivalent to ∑v∈V LG,v; whence N = TSN ≅ LG,V . ′ ′ With this in hand, we can prove an analogue of [16, Theorem 2.7] relating intertwiners and absolute continuity.

Theorem 7.5. Let S = (Sv,Se) be a TCK family on H for a graph G corresponding to a representation πS of T (G). Then

(1) ψ(A) = ⟨πS(A)x, y⟩ is absolutely continuous for all x, y ∈ Vac. (2) Vac is a closed subspace consisting of all absolutely continuous vectors. (3) Vac is invariant for S. (4) Vac ⊕ 0 lies in the closed span of the wandering vectors for S ⊕ L in H ⊕ HG.

Proof. Since x, y are in Vac, there are maps X,Y ∈ X (S) and ζ, η ∈ HG such that x = Xζ and y = Y η. Therefore ∗ ψ(A) = ⟨πS(A)Xζ, Y η⟩ = ⟨XπL(A)ζ, Y η⟩ = ⟨πL(A)ζ, X Y η⟩. Hence ψ is absolutely continuous, and (1) holds. In particular, every vector in Vac is absolutely continuous. If x is absolutely continuous, then by Lemma 7.4 there is a vector ζ ∈ HG so that N = T[x ⊕ ζ] ≅ HG,V . This means that there is a partial isometry U ∈ B(HG, H) with domain H′ G,V and range N so that ′ + (Sµ ⊕ Lµ)U = ULµ for all µ ∈ F (G).

Let X = PHU and observe that this is an intertwiner in X (S). There is a ∗ vector ξ = U (x ⊕ ζ) ∈ HG such that Uξ = x ⊕ ζ. Therefore Xξ = x belongs to Vac. Combining this with (1) shows that Vac consists of all absolutely continuous vectors. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 37

It now follows that Vac is a closed subspace. Indeed, if x, y ∈ Vac, then clearly so are scalar multiples. Moreover

⟨πS(A)x+y, x+y⟩ = ⟨πS(A)x, x⟩ + ⟨πS(A)x, y⟩ + ⟨πS(A)y, x⟩ + ⟨πS(A)y, y⟩ is a sum of weak-∗ functionals on LG, and hence is a weak-∗ continuous. So x + y is absolutely continuous. Similarly Vac is closed because the predual of LG is complete. + If x ∈ Vac and A ∈ T (G), select a ξ ∈ HG and X ∈ X (S) so that x = Xξ. Then πS(A)x = πS(A)Xξ = XπL(A)ξ + is in the range of X. Thus Vac is invariant for πS(T (G)), and hence is also invariant for its wot closure S. Finally returning to the setup in the second paragraph, we see that x ⊕ ζ lies in a subspace of left regular type, and hence lies in the closed span of wandering vectors. By Lemma 7.4, this can be accomplished with YζY < ε, and thus x ⊕ 0 also lies in the closed span of wandering vectors.

Corollary 7.6. If S is a TCK family for G, then Vac is the unique largest invariant subspace K such that SSK is absolutely continuous.

Proof. By Theorem 7.5 Vac is invariant and SSVac is absolutely continuous. Also by deﬁnition of absolute continuity and Theorem 7.5 again, we see that if K is any invariant subspace for which SSK is absolutely continuous, then K ⊆ Vac. Our next goal is to understand the precise relationship between absolutely continuous and analytic free semigroupoid algebras. Corollary 7.7. Let G be a graph and S a free semigroupoid algebra generated by a TCK family S. If S is analytic, then it is absolutely continuous. Proof. Each wandering vector x is absolutely continuous because S[x] ≅ HG,supp x, and this unitary equivalence is an intertwining operator with range S[x]. Thus the span of wandering vectors lies in Vac. By item (6) of The- orem 6.15, it follows that H is the closed linear span of wandering vectors. Therefore Vac = H, and S is absolutely continuous.

Example 7.8. Let G be a graph which contains a cycle Cn that has no entries; i.e. the cycle has edges e1, . . . , en on vertices v1, . . . , vn and if f ∈ E such that r(f) = vi+1(mod n) for some 1 ≤ i ≤ n, then f = ei. Let A be a measurable subset of the circle T such that 0 < m(A) < 1. Define a repre- + n 2 sentation π of T (Cn) on V = C ⊗ L (A) by setting π(ei) = Ei,i+1 ⊗ I for 1 ≤ i < n and π(en) = En,1 ⊗ Mz,A, where Mz,A is multiplication by z on L2(A). Now enlarge the space and extend this in any manner to a represen- + tation π1 of T (G) by defining the images of the other generators. There are various ways to do this, and we leave the details to the reader. Then restrict to the cyclic subspace S[V]. This determines a free semigroupoid algebra S1 for G. This extension is now unique up to unitary equivalence 38 K.R. DAVIDSON, A. DOR-ON, AND B. LI because the Wold Decomposition shows that W = ∑⊕ S V ⊖ V is a s(e)∈Cn e ⊥ wandering subspace generating V . Let π2 be a second representation using c A = T ∖ A instead of A, and S2 its free semigroupoid algebra. We now apply Theorem 5.6 about representations of a cycle. The rep- n 2 2 c resentation π1 ⊕ π2 compressed to C ⊗ (L (A) ⊕ L (A )) is spanned by wandering vectors, and is thus is analytic. Therefore both π1 and π2 are n 2 n 2 c absolutely continuous on C ⊗ L (A) and C ⊗ L (A ) respectively. How- ever S1 and S2 are of von Neumann algebras, and are hence not analytic. Thus, S1 and S2 are absolutely continuous free semigroupoid algebras that are not analytic. It turns out that this example is the only possible pathology. Note that if µ = en . . . e1 is a cycle in a graph G and Sµ is a unitary map on Ss(µ)H, then any edge f distinct from en, ..., e1 with r(f) ∈ {v1, . . . , vn} must have Sf = 0. Conversely, for a CK family with this property, we must have that Sµ unitary on Ss(µ)H. If the representation is absolutely continuous, then the spectral measure of Sµ must be absolutely continuous with respect to Lebesgue measure on T. As the example above shows, in order to be analytic, the spectral measure must be equivalent to Lebesgue measure. Equivalently, we could assume that the spectral measure dominates Lebesgue measure on T and is absolutely continuous. By analogy to the case of free semigroup algebras, we adopt the following definition from [16]. Definition 7.9. A free semigroupoid algebra S is regular if the absolutely continuous part is analytic. We note that a free semigroup algebra S can fail to be regular only if the free semigroup has a single generator, and the isometry generating it is unitary and the spectral measure has a proper absolutely continuous part which is not equivalent to m. This follows from Wermer’s Theorem as in Example 7.8. When this notion was introduced in [16], it was not clear what could happen when the number of generators is two or more. Kennedy [36] showed that free semigroup algebras on at least two generators are always regular. The following characterizes regular free semigroupoid algebras. Theorem 7.10. Suppose that S is a free semigroupoid algebra of a graph G generated by a TCK family S on H. Then S is regular if and only if it satifies the measure theoretic condition

(M) Whenever µ is a cycle such that Sµ is unitary on Ss(µ)H, the spectral measure of Sµ is either singular or dominates Lebesgue on T. Proof. First assume that (M) fails to hold. Then there is a cycle µ such that Sµ is unitary, say with decomposition Sµ ≅ Ua ⊕Us on Ss(µ)H ≅ Na ⊕Ns into its absolutely continuous and singular parts, such that Ua ≠ 0 is supported on a subset A ⊆ T with 0 < m(A) < 1. Since Sµ is unitary, the cycle µ has no non-zero entries; i.e., in the graph G[supp S], the cycle µ has no entries. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 39

By the analysis in Theorem 5.6 and Example 7.8, we see that the subspace Na is absolutely continuous but contains no wandering vectors, and thus is not analytic. Hence S is not regular. Now suppose that (M) holds. By Proposition 7.7, the analytic part is absolutely continuous. For the converse, by restricting to the absolutely continuous part, we may assume that S is absolutely continuous, so that H = Vac. By Theorem 6.15(6), we need to show that H is spanned by wandering vectors. By Proposition 5.1 and Proposition 5.3, it is enough to show that each SvH is spanned by wandering vectors for the free semigroup algebra

Sv = SvSSSvH. The free semigroup is generated by the irreducible cycles at v. Since S is absolutely continuous, each Sv is absolutely continuous. There are then three cases. If there are no cycles through v, then Sv ≅ C and every vector in SvH is wandering. If there are at least 2 irreducible cycle operators, then by [36, Theorem 4.16], Sv is analytic type and H is spanned by wandering vectors. Finally if v lies on a single irreducible cycle µ, Theorem 5.6 shows that Sv is generated by Sµ. If Sµ is not unitary, then its Wold decomposition contains copies of the unilateral shift U+. As S is absolutely continuous, this falls into case (2) of Theorem 5.6 and the spectral measure of the unitary part is absolutely continuous. If Sµ is unitary, then by hypothesis, the spectral measure of Sµ dominates m, which is also case (2) of Theorem 5.6. Hence, in either case, the wandering vectors span the absolutely continuous part (i.e. all of SvH).

Corollary 7.11. Let S and T be TCK families for G on HS and HT respectively, and assume that S is regular. Then S ⊕ T is regular if and only if whenever µ is a cycle and Tµ is unitary on Ts(µ)HT, we either have that (1) the spectral measure of Tµ is singular or dominates Lebesgue on T; or ⊥ (2) PS HS,s(µ) ≠ {0}, where PS is the structure projection for S. In particular, L ⊕ T is always regular. Proof. Since S is regular, it always satisﬁes (M). So if (1) holds, then (M) is satisﬁed for µ. On the other hand, if (2) holds, then S has wandering vectors in HS,s(µ) by the Structure Theorem 6.15 (6). It follows that Sµ acts as a shift on the cyclic subspace generated by the wandering vector, and hence Sµ is either a proper isometry or contains a summand unitarily equivalent to the bilateral shift, in which case its spectral measure dominates Lebesgue measure. Either way, (M) holds. So S ⊕ T is regular. Now L has analytic type, and is supported on every vertex, so (2) holds generally. Thus L ⊕ T is always regular.

Example 7.12. Let S = (Sv,Se) be an arbitrary TCK family on H. Let U 2 be the bilateral shift on ` (Z). Then the TCK family T = (Sv ⊗ I,Se ⊗ U) 2 always generates an analytic free semigroupoid algebra T on H ⊗ ` (Z). 2 To see this, let {ξn ∶ n ∈ Z} be the standard basis for ` (Z). Fix any non-zero vector in x ∈ H. We claim that x ⊗ ξn is a wandering vector. Note 40 K.R. DAVIDSON, A. DOR-ON, AND B. LI that if SµS = k, then k Tµ(x ⊗ ξn) = Sµ ⊗ U (x ⊗ ξn) = Sµx ⊗ ξn+k.

Therefore the subspace T[x ⊗ ξn] is graded, and hence if SµS = k ≠ l = SνS, then ⟨Tµ(x ⊗ ξn),Tν(x ⊗ ξn)⟩ = ⟨Sµx ⊗ ξn+k,Sνx ⊗ ξn+l⟩ = 0. Also if SµS = SνS, then Sµ and Sν have pairwise orthogonal ranges; so that

⟨Tµ(x ⊗ ξn),Tν(x ⊗ ξn)⟩ = ⟨Sµx, Sνx⟩ = 0.

Thus x ⊗ ξn is a wandering vector. This shows that the wandering vectors 2 span H ⊗ ` (Z), and therefore T is analytic type. 8. Lebesgue-von Neumann-Wold decomposition We apply the Structure Theorem to obtain a decomposition theorem for regular Toeplitz-Cuntz-Krieger families. We show that the classification of regular representations of the Toeplitz-Cuntz-Krieger algebra T (G) reduces to the classification of fully coisometric absolutely continuous, singular and dilation type representations. This generalizes Kennedy’s Lebesgue decomposition for free semigroup algebras [36, Theorem 6.5]. We already have left regular and analytic TCK families and now absolutely continuous ones. We define a few other types.

Deﬁnition 8.1. Let S = (Sv,Se) be an TCK family for a graph G. We will say that S is (1) singular if it has no absolutely continuous restriction to any invariant subspace. (2) von Neumann type if S is a von Neumann algebra. (3) dilation type if it has no restriction to a reducing subspace that is either analytic or a von Neumann algebra. Note that if a free semigroupoid algebra S is singular, then the structure projection P = I since the range of P ⊥ is analytic and hence absolutely continuous. Therefore S is a von Neumann algebra. But we know from Theorem 5.6 that there are absolutely continuous free semigroupoid algebras which von Neumann algebras. However these examples are not regular. The following is standard, but we include it for the reader’s convenience. Lemma 8.2. Suppose that S is a free semigroupoid algebra and that V is a coinvariant subspace. Then S[V] and S∗[V⊥] are reducing for S. Proof. The proofs of the two statements are similar, and we prove only the ﬁrst. Observe that S[V] is invariant for S by construction, and it is + spanned by vectors of the form Sµξ for ξ ∈ V and µ ∈ F (G). It follows that for e ∈ E, ⎧ = ′ ⎪Sµ ξ if µ eµ ∗ ⎪ S S ξ = ⎨0′ if µ = fµ′ and f ≠ e e µ ⎪ ⎪ ∗ ⎩⎪Se ξ if µ = ∅. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 41

∗ ∗ The co-invariance of V shows that Se ξ ∈ V; and therefore Se S[V] ⊆ S[V]. Thus S[V] is reducing.

Lemma 8.3. Let G be a graph, and let S = (Sv,Se) be a TCK family for G. Let P be the structure projection for S, and set V = P H. Then S is of dilation type if and only if V is cyclic for S and V⊥ is cyclic for S∗. If S is regular, then S is of dilation type if and only if it does not have an absolutely continuous summand or a singular summand. Let A ∶= P SSP H. Then A is a contractive G family and S is the unique minimal dilation of A to a TCK family. Moreover, the restriction SSP H is unitarily equivalent to a left regular family. ⊥ Proof. Assume that S is dilation type, and let V = P H. Since V is coinvariant, the subspaces S[V] and S∗[V⊥] reduce S by Lemma 8.2. Now S[V]⊥ is a reducing subspace contained in V⊥, and hence it is an analytic, and thus an absolutely continuous reducing subspace. So it must be {0}, and V is cyclic. Similarly, N = S∗[V⊥]⊥ is a reducing subspace for S contained in V, and thus SSN is a von Neumann algebra. Again this means that it is {0}. Therefore V⊥ is cyclic for S∗. Conversely, if V is cyclic, then there is no reducing subspace in V⊥; so there is no analytic summand. Similarly if V⊥ is cyclic for S∗, then there is no reducing subspace of V, which would be of von Neumann type. Thus S is dilation type. We need also to verify that if S is regular, then it has no absolutely continuous summand. However in the regular case, the absolutely continuous part is spanned by wandering vectors and lives in the analytic part. Thus any absolutely continuous summand is an analytic summand, which cannot occur. See Example 8.4 to explain what happens when S is not regular. Since P H is coinvariant for S, the compression of πS to P H is a completely contractive homomorphism, which we restrict to T+(G). This representation is determined by the image of the generators, namely A = {Av,Ae}. By a result of Muhly and Solel [39, Theorem 3.3], A has a unique minimal dilation to a TCK family. As S is also a minimal dilation of A by the previous paragraph, it is the unique minimal dilation up to a unitary equivalence ﬁxing V. ⊕ Finally, let W = V + ∑e∈E SeV ⊖ V. It is readily veriﬁed that this is a non-empty wandering subspace for SSV and that H = S[V] = V⊥ ⊕ S[W].

Therefore W is a cyclic wandering subspace for SSV , which then must be left regular type by the Wold Decomposition 4.2. ⊥ Remark 8.4. Regularity is required for the second statement in this lemma. Consider the graph G on two vertices v, w with edges e, f, g such that s(e) = r(e) = s(f) = v and r(f) = s(g) = r(g) = w. Consider the representation π1 coming from the cycle at v, and the corresponding CK family S of G constructed for this graph as in Example 7.8. This representation is 42 K.R. DAVIDSON, A. DOR-ON, AND B. LI

⊥ absolutely continuous. However V = SvH is cyclic and V = SwH is cyclic for S∗. Hence S is absolutely continuous and of dilation type. We are now able to provide the analogue of Kennedy’s Lebesgue decomposition for isometric tuples [36, Theorem 6.5]. Theorem 8.5. Let G be a graph and let S be a TCK family on H generating a free semigroupoid algebra S. Then up to unitary equivalence we may decompose S ≅ Sl ⊕ San ⊕ SvN ⊕ Sd where

(1)S l is unitarily equivalent to a left-regular TCK family. (2)S an is an analytic fully coisometric CK family. (3)S vN is a von Neumann type fully coisometric CK family. (4)S d is a dilation type fully coisometric CK family. If S is regular, then the absolutely continuous part coincides with the analytic part, and so San coincides with the largest absolutely continuous fully coisometric CK summand Sa. Moreover, the largest singular summand Ss coincides with SvN . Thus our decomposition can be written instead as

S ≅ Sl ⊕ Sa ⊕ Ss ⊕ Sd.

Proof. By the Wold-decomposition, we may write S = Sl ⊕ U with respect ′ to a decomposition H = Hl ⊕ H , where Sl is a left-regular TCK family, and U is a fully coisometric CK family on H′. Let P be the structure projection for S; and let V ∶= P H. Note that since Sl is analytic, V is orthogonal to Hl. Deﬁne reducing subspaces M = S[V] and N = S∗[V⊥]. ⊥ Since Hl is a reducing subspace contained in V , we see that M is orthogonal to Hl. Deﬁne ⊥ ⊥ Han = (N ∩ M ) ⊖ Hl, HvN = M ∩ N and Hd = M ∩ N .

Set San = SSHan ,SvN = SSHvN and Sd = SSHd . ⊥ Because M ⊇ V and N ⊇ V , it follows that the projections PM and PN commute with each other and with P . We compute that M⊥ ∩ N ⊥ ⊆ V⊥ ∩ V = {0}. Therefore, H = (N ∩ M⊥) ⊕ (M ∩ N ⊥) ⊕ (M ∩ N )

= (Hl ⊕ Han) ⊕ HvN ⊕ Hd. ⊥ Note that Han is contained in V , and hence San has analytic type. As the left regular summand has been removed from Han, what remains is a fully coisometric CK family. Next observe that HvN is contained in V, and therefore SvN is von Neumann type. To verify that Sd has dilation type, ⊥ we apply Lemma 8.3. Let V1 = P Hd and W1 = P Hd. Then Hd = V1 ⊕ W1 STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 43 because PM and PN commute with P . Because V ⊖ V1 = HvN is reducing ⊥ and likewise V ⊖ W1 = Han is reducing, we obtain ⊥ ⊥ ∗ ∗ ⊥ S[V1] ∩ V = S[V] ∩ V = W1 and S [W1] ∩ V = S [V ] ∩ V = V1.

Therefore Hd is dilation type. Now suppose that S is regular. By deﬁnition, the absolutely continuous ⊥ subspace Vac coincides with P H. Thus the largest absolutely continuous summand coincides with the largest analytic summand, namely Sl ⊕San. So the largest fully coisometric CK absolutely continuous summand Sa is equal to San. Also the von Neumann part is contained in V and thus is singular; so Ss = SvN . Hence we obtain the decomposition as in the second part of the statement. Proposition 8.6. Let G be a graph, and let S, T be TCK families generating a free semigroupoid algebras S, T on HS, HT respectively. The summands of a free semigroupoid algebra of left regular, absolutely continuous and singular type are intrinsic in the sense that

Sl ⊕ Tl = (S ⊕ T)l, Sa ⊕ Ta = (S ⊕ T)a and Ss ⊕ Ts = (S ⊕ T)s. Furthermore, if S is regular, then the summands of analytic, von Neumann and dilation type are unchanged by adding a direct summand to S in the sense that

San = (S ⊕ T)anSHS , SvN = (S ⊕ T)vN SHS and Sd = (S ⊕ T)dSHS . Finally, if both S and T are regular, then S ⊕ T is regular and

San ⊕ Tan = (S ⊕ T)an and SvN ⊕ TvN = (S ⊕ T)vN . Proof. The left regular part is determined by the Wold decomposition, and depends on the complement of the ranges of edges with range v. A direct sum just produces the direct sum of the left-regular spaces, which is the left- regular part. The absolutely continuous part is supported on the subspace Vac of absolutely continuous vectors, which by Corollary 7.6 is preserved under direct sums. Thus it is unaﬀected by other summands. Therefore the complementary singular part is also intrinsic. Next, when S is regular, we see that (S ⊕ T)a = Sa ⊕ Ta = San ⊕ Ta, and clearly the analytic part of S⊕T includes San as a summand and is contained in the absolutely continuous part of S ⊕ T. Hence, (S ⊕ T)anSHS = San. It similarly follows that the same holds for von Neumann and dilation types. Finally, when both S and T are regular, it is clear from condition (M) of Theorem 7.10 that so is S ⊕ T, and the rest follows from the ﬁrst part.

9. Applications In this section we provide three applications of the theory we have developed. We show that every free semigroupoid algebra is reflexive, that every 44 K.R. DAVIDSON, A. DOR-ON, AND B. LI regular free semigroupoid algebra satisfies a Kaplansky-type density theorem, and that isomorphisms of nonself-adjoint free semigroupoid algebras can recover any transitive, row-finite graph from their isomorphism class. First, we apply our structure theorem to obtain reflexivity of free semigroupoid algebras. Recall that a subalgebra A ⊆ B(H) is reflexive if A = Alg Lat A = { T ∈ B(H)S P ⊥TP = 0 for every P ∈ Lat A } where Lat A is the lattice of closed invariant subspaces for A ⊆ B(H). Kribs and Power [37, Theorem 7.1] show that the left regular free semigroupoid algebra LG ⊆ B(HG) is reflexive for every directed graph G. A careful look at their proof shows that they actually show that LG,v ⊆ B(HG,v) is reflexive for any vertex v. By taking direct sums, we see that every left regular type free semigroupoid algebra is reflexive. We now leverage this result to obtain reflexivity of free semigroupoid algebras. Theorem 9.1. Let S be a free semigroupoid algebra for a graph G generated by a TCK family S = (Sv,Se) on H. Then S ⊆ B(H) is reflexive. Proof. By the double commutant theorem, every von Neumann algebra is reflexive. Thus Alg Lat S ⊆ Alg Lat M = M. Let P be the structure projection of S from Theorem 6.15. Since P ⊥H is invariant for S, it is also invariant for any T ∈ Alg Lat S. Thus we may write T = TP + P ⊥TP ⊥, where T ∈ M. The Structure Theorem also shows that TP ∈ MP = SP ⊆ S. Hence the problem reduces to showing that the analytic part, SSP H, is reflexive. So we may suppose that P = 0 and S is analytic. The key⊥ to our proof is the fact that H is spanned by wandering vectors, another consequence of the Structure Theorem. Observe that since Pv ∈ S and T = wot–∑v∈V TPv, it suffices to show that each TPv belongs to S. Denote Hw = PwH for each w ∈ V . Next, fix a vertex v ∈ V such that Pv ≠ 0, and note that Hv is spanned by wandering vectors, since if x is wandering, so is Pvx. Let G[v] be the directed subgraph generated by v on the vertices Vv. Then there is no harm is restricting our attention to SS where K = ∑⊕ H . For if TP S belongs to Alg Lat SS , K w∈Vv w v K K then a fortiori, TPv belongs to Alg Lat S. Because S is analytic, we have a canonical isomorphism ϕ ∶ LG,v → SSK. Let x0 be a fixed wandering vector in Hv, and let x1 be any other wandering vector in Hv. Let Mi = S[xi], and let Vi ∶ HG,v → Mi be the canonical ∗ isometry given by Viξµ = Sµxi. Then ϕi(A) = ViAVi is a unitary equivalence of LG,v onto SSMi , and moreover ϕi(A) = ϕ(A)SMi . Now LG,v is reflexive by [37, Theorem 7.1] (and the observation preceding this proof). Thus TPv leaves Mi invariant, and TPvSMi ∈ Alg Lat SSMi = SSMi . Therefore there are operators Ai ∈ LG,v so that TPvSMi = ϕi(Ai). By replacing TPv by TPv − ϕ(A0), we may also suppose that A0 = 0. If we can show that A1 = 0 for any wandering vector x1, then it follows that TPv = 0, and hence it lies in S. Now A1 = A1Lv has a Fourier series determined by the coefficients of STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 45

η = A1ξv; and if η = 0, then we would obtain that A1 = 0. Since V1 is an isometry, this is further reduced to showing that y ∶= V1η = 0. Let Vt = V0 +tV1 for 0 < t ≤ 0.5. Observe that ViA = ϕ(A)Vi for i = 0, 1 and A ∈ LG,v, and therefore VtA = ϕ(A)Vt holds as well; i.e. Vt are intertwining maps. Moreover, the operator Vt is bounded below by 1−t ≥ .5, and therefore has closed range Mt = VtHG,v. The fact that Vt intertwines S and LG,v shows that Mt is invariant for S, and hence for TPv. Let xt = Vtξv = x0 +tx1, and note that

y = V1η = V1A1ξv = ϕ1(A1)V1ξv = T x1.

Then, y ∈ M1 and since T SM0 = 0, we have that Mt contains

TPvxt = T x0 + tT x1 = ty.

Thus y lies in M0. To see this note that there are vectors ζt ∈ HG,v such −1 that Vtζt = y. As Vt is bounded below by 1 − t, we have YζtY ≤ (1 − t) ≤ 2. Thus for 0 < s < t < 1,

Yζs − ζtY = YVt(ζs − ζt)Y = Y(Vt − Vs)ζsY = (t − s)YζsY ≤ 2(t − s).

So ζt is Cauchy as t → 0 with limit ζ0, and

y = lim Vtζt = V0ζ0 ∈ M0. t→0

Therefore y ∈ M0 ∩ M1. We wish to show that y = 0, and there are three cases depending on how many irreducible cycles there are at v. If v does not lie on any cycles, then every vector in Hv is wandering. We may select an orthonormal basis {xk ∶ 0 ≤ k < dim Hv} for Hv. Then S[xk] are pairwise orthogonal. Since ∈ [ ] ∩ [ ] = { } ≥ S = y S x0 S xk 0 for each xk, k 1;, we obtain that TPv S[xk] 0. ⊕ Since K = ∑k≥0 S[xi], we deduce that TPv belongs to S. + If v lies on two or more irreducible cycles, say µ1 and µ2 in F (G)v, then x0i = Sµi x0 are also wandering vectors such that M0i ∶= S[x0i] are pairwise orthogonal subspaces of M0. Thus we have that TPvSM0i = 0. The same analysis as above can be repeated for both of these invariant subspaces. We deduce that y ∈ M01 ∩ M02 = {0}. Again we see that TPv belongs to S. The final case occurs when there is exactly one irreducible cycle µ at v. Here we use the fact that S is analytic to see that Sµ is unitarily equivalent (α) to U+ ⊕U where U is a unitary with spectral measure absolutely continuous to Lebesgue measure m, and if α = 0, then the spectral measure is equivalent to m (Theorem 5.6). In the latter case, U has a reducing subspace on which it is unitarily equivalent to the bilateral shift, and hence has an invariant subspace on which it is unitarily equivalent to the unilateral shift U+. So in either case, we have a subspace M0 of Hv with an orthonormal basis yk for k ≥ 0 such that Sµyk = yk+1. Observe that each yk is a wandering vector for Sv, as this is the algebra generated by SµSHv . By Lemma 5.3, the yk are also wandering vectors for S. Now we take x0 = y0. As in the previous paragraph, we observe that we could use yk in lieu of y0. It follows that 46 K.R. DAVIDSON, A. DOR-ON, AND B. LI y lies in ⋂k≥0 S[yk] = {0}. Again we find that TPx = 0 ∈ S. Thus S is reflexive. The astute reader will wonder about hyper-reflexivity. The analysis of this issue is more involved, and we are leaving this topic to another paper. Our next goal is to establish an analogue of the Kaplansky density theorem akin to [16, Theorem 5.4] for regular free semigroupoid algebras, or alternatively those free semigroupoid algebras whose TCK families satisfying condition (M). Theorem 9.2. Let G be a graph, and let S be a regular TCK family generating the free semigroupoid algebra S. Let πS be the corresponding ∗- representation of T (G). Then the unit ball of Mn(πS(T+(G))) is weak-∗ dense in the unit ball of Mn(S).

Proof. Fix a separable Hilbert space H, and let πu denote the direct sum of all ∗-representations of T (G) on H. Then the universal von Neumann ′′ algebra Mu = πu(T (G)) is canonically ∗-isomorphic and weak-∗ homeomorphic to T (G)∗∗. It is a standard result that every representation π of T (G) extends uniquely to a weak-∗ continuous ∗-representationπ ¯ of Mu, ′′ and thatπ ¯(Mu) = π(T (G)) . Moreover there is a central projection Qπ ⊥ so that kerπ ¯ = MuQπ. The representation πu is unitarily equivalent to its inﬁnite ampliation. Hence the wot and weak-∗ topologies coincide on Mu. Since T+(G) is a closed subalgebra of T (G), basic functional analysis ∗∗ shows that T+(G) may be naturally identiﬁed with the weak-∗ closure Su of πu(T+(G)) in Mu. This may be considered to be the universal free semigroupoid algebra for G. We apply the Structure Theorem 6.15 to Su to obtain the structure projection Pu so that ⊥ ⊥ ⊥ ⊥ ⊥ Su = MuPu + Pu SuPu and Pu SuPu = SuPu ≅ LG.

By Goldstine’s Theorem, the unit ball of T+(G) is weak-∗ dense in the unit ∗∗ ball of T+(G) ; whence the unit ball of πu(T+(G)) is weak-∗ dense in the unit ball of Su. Letπ ˜S be the restriction ofπ ¯S to Su. Becauseπ ˜S is weak-∗ continuous, it will map Su into S. Let P be the structure projection for S. We wish to show that P = π¯(PuQπS ) = π¯(Pu). From the Structure Theorem 6.15(6), ⊥ ⊥ we know that both Pu Hu and P HπS are spanned by wandering vectors, ⊥ ⊥ so thatπ ¯S(Pu ) ≥ P . On the other hand, by Proposition 7.7 we know that ⊥ the analytic part Pu Hu of Su is absolutely continuous. Hence, the subspace ⊥ ⊥ Pu HπS is absolutely continuous for S and contains the analytic part P HπS . ⊥ ⊥ Since S is regular, these spaces coincide, so thatπ ¯S(Pu ) = P . If S is a von Neumann algebra, then the structure projection is P = I. ⊥ ⊥ Sinceπ ¯(Pu QπS ) = P = 0, we see that QπS ≤ Pu. Thus QπS ∈ MuPu, and hence belongs to Su. The map taking X ∈ Su to XQπS ∈ MuQπS = S is clearly a normal complete quotient map. ⊥ ⊥ Otherwise P ≠ 0. The restriction ofπ ˜S to the corner SuPu factors as ⊥ ⊥ SuPu ≅ LG → LG[Vw] ≅ SP , STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 47

where the map in the middle is the complete quotient map ρV,Vw of Theo- rem 3.1, where Vw are the vertices v such that Sv ∉ S0, as in the Structure Theorem 6.15. In particular, this map is surjective. Furthermore, the restriction ofπ ˜S to the left ideal MuPu is given by multiplication by QS and maps onto MP . Combining these two together we obtain thatπ ˜S maps Su onto S. ⊥ For X ∈ S, we write X = XP + XP . As P = π¯S(PuQS), we get that MP is canonically ∗-isomorphic to MuPuQS. Thus, there is an element ⊥ Y1 = Y1PuQS ∈ Mu such thatπ ¯S(Y1) = XP . The corner XP is identiﬁed with an element A ∈ LG[Vw]. Let B = jVw,V (A) be the image of A in LG via the completely isometric section jVw,V as in Theorem 3.1. Let Y2 be the ⊥ element of SuPu which is identiﬁed with B. Then Y = Y1 + Y2 ∈ Su and ⊥ X = π˜S(Y ) = YQS. Moreover by construction, Y − YQS = Y2QS. Therefore, since QS is central, ⊥ YY Y = max{YYQSY, YYQSY} = max{YXY, YBY} = max{YXY, YAY} = YXY.

It follows thatπ ˜S is a quotient map of Su onto S. The identical argument works for matrices over Su, and thusπ ˜S is a complete quotient map. Finally, let X belong to the unit ball of S. Pick a Y in the unit ball of Su such thatπ ¯S(Y ) = X. Using Goldstine’s Theorem, select a net Aα in the unit ball of T+(G) so that πu(Aα) converges weak-∗ to Y . Then π¯Sπu(Aα) = πS(Aα) is a net in the unit ball of πS(T+(G)) which converges weak-∗ to X. The same argument then works for any matrix. Example 9.3. This theorem is false for some free semigroupoid algebras of a cycle. For simplicity, consider the graph C1 on a single vertex v with one edge e from v to itself. Then T+(C1) is naturally isomorphic to A(D). Fix a subset A ⊂ T with 0 < m(A) < 1, and consider the representation πA on 2 L (A) sending Se to Mz,A as in Example 7.8. Then the free semigroupoid ∞ algebra is L (A). However the unit ball of πA(A(D)) cannot be weak-∗ dense in the unit ball of L∞(A). Indeed, if this were true, then for each ∞ f ∈ L (A) of norm 1, there would be a sequence hn in the unit ball of A(D) such that πA(hn) converges weak-∗ to Mf . In particular, hn converges to f m a.e. on A. Considering this as a sequence in the unit ball of H∞, there would be a subsequence which also converges weak-∗ to some h in the unit ∞ 2 ball of H . This would mean that in B(L (T)) that Mhn converges weak-∗ to Mh, and in particular hn converges to h a.e.(m) on T. By restriction to A, ∞ we can conclude that f = hSA. But not every f ∈ L (A) is the restriction of an H∞ function, let alone one of the same norm. As an example we can take ∞ B ⊆ A with 0 < m(B) < m(A), and deﬁne f = χB. If there was h ∈ H such ∞ that hSA = χB, we see that h is a non-zero function in H which vanishes on a subset of T with non-zero measure. This contradicts the F. and M. Riesz Theorem (see [22, Theorem 6.13]). 48 K.R. DAVIDSON, A. DOR-ON, AND B. LI

Finally, we conclude this section by obtaining some isomorphism theorems for nonself-adjoint free semigroupoid algebras of transitive row-ﬁnite graphs. Proposition 9.4. Let S be a free semigroupoid algebra for a transitive graph ∗ G generated by a TCK family S = (Sv,Se) on H. Let M = W (S) be the von Neumann algebra generated by S, and let P be the structure projection given in Theorem 6.15. Then the Jacobson radical of S is P ⊥SP . Proof. Clearly P ⊥SP is a nil ideal, and is hence contained in the Jacobson radical. On the other hand, by [37, Theorem 5.1] LG is semisimple because G is transitive. Thus the analytic type part SP ⊥ is semisimple. Therefore the quotient S~P ⊥SP ≅ P MP ⊕ SP ⊥ is a direct sum of semisimple operator algebras, and hence is semisimple. Thus P ⊥SP is the Jacobson radical of S. Lemma 9.5. Let G be a countable directed graph. Then every idempotent 0 ≠ P ∈ LG must be ﬁnite. That is, if Q ∈ LG is an idempotent such that PQ = QP = Q and P and Q are algebraically equivalent. Then P = Q. ∞ Proof. Let Φ0 ∶ LG → ` (V ) be the canonical conditional expectation, which is also a homomorphism. Then for every non-zero idempotent Q ∈ LG, k k we have that Φ0(Q) ≠ 0. Indeed, if not, then Q = Q ∈ LG,0 for all k ≥ 0 and hence must be 0. Next, suppose that A, B ∈ LG satisfy AB = P and BA = Q. Since Φ0 is a homomorphism of LG into a commutative algebra, we see that

Φ0(P ) = Φ0(A)Φ0(B) = Φ0(B)Φ0(A) = Φ0(Q).

Hence Φ0(P − Q) = 0. Our assumptions guarantee that P − Q is an idempotent, and therefore P = Q by the ﬁrst paragraph. Next we will show that when the graphs are transitive, the structure projection P appearing in Theorem 6.15 is intrinsic to the free semigroupoid algebra. This will allow us to recover the underlying graph from an algebraic isomorphism class.

Theorem 9.6. Let S1 and S2 be nonself-adjoint free semigroupoid algebras for transitive graphs G1 and G2. Suppose that ϕ ∶ S1 → S2 is an algebraic isomorphism. Let P1 and P2 be the structure projections of S1 and S2 as in Theorem 6.15. Then there is an invertible element V ∈ S2 such that ϕˆ ∶= AdV ○ϕ satisﬁes ϕˆ(P1) = P2. Hence, there is an algebraic isomorphism between LG1 and LG2 . ⊥ ⊥ Proof. Since both S1 and S2 are nonself-adjoint, we see that P1 and P2 are both non-zero. Let N1 = P1S1P1 and N2 = P2S2P2 be the von Neumann algebra corners of the free semigroupoid algebras, as in Theorem 6.15. The isomorphism must carry the Jacobson radical of S1 onto the radical of S2. Therefore there is an induced isomorphism ϕ̃ taking the quotient by the

Jacobson radical, which is isomorphic to N1 ⊕ LG1 , onto N2 ⊕ LG2 . In STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 49

⊥ particular, this carries the centre to the centre, so ϕ̃ carries Z(N1) ⊕ CP1 ⊥ onto Z(N2) ⊕ CP2 . ⊥ ⊥ Note that Pi is a minimal projection in the respective centers. So ϕ̃(P1 ) ⊥ ⊥ is a minimal projection in Z(N2) ⊕ CP2 . If this is not P2 , then it is a minimal projection Q ∈ Z(N2). However this would imply an isomorphism between LG1 and a von Neumann algebra QN2Q. Since G1 is transitive,

LG1 is inﬁnite dimensional, and the ideal LG1,0 properly contains its square L2 . However every ideal of any C*-algebra is equal to its own square; so G1,0 ⊥ ⊥ LG1 is not isomorphic to a von Neumann algebra. Hence ϕ̃(P1 ) = P2 . ⊥ ⊥ ⊥ This means that ϕ(P1 ) = P2 + X where X = P2 XP2 lies in the Jacobson 2 radical of S2 and satisﬁes X = 0. Hence, taking V = I + X, we see that −1 ⊥ −1 ⊥ V = I − X and V ϕ(P1 )V = P2 . The algebraic isomorphismϕ ˆ ∶ S1 → S2 −1 ⊥ ⊥ given byϕ ˆ(A) = V ϕ(A)V maps P1 to P2 , and thusϕ ˆ(P1) = P2. ⊥ Finally, since SiPi is isomorphic to LGi for i = 1, 2, this yields an algebraic isomorphism between LG1 and LG2 .

Corollary 9.7. Let G1 and G2 be transitive row-ﬁnite graphs. The following are equivalent

(1) G1 is isomorphic to G2.

(2) LG1 and LG2 are completely isometrically isomorphic and weak-∗ homeomorphic. (3) There exist two algebraically isomorphic nonself-adjoint free semigroupoid algebras S1 and S2 for G1 and G2 respectively.

Proof. Clearly if G1 and G2 are isomorphic, then LG1 and LG2 are completely isometrically isomorphic and weak-∗ homeomorphic. So (1) implies

(2). LG1 and LG2 are examples of nonself-adjoint free semigroupoid algebras, so (2) implies (3). Finally, by Theorem 9.6, an algebraic isomorphism between S1 and S2 yields an algebraic isomorphism between LG1 and LG2 . By [33, Corollary 3.17], G1 and G2 are isomorphic. So (3) implies (1). Corollary 9.8. Let S be a free semigroupoid algebra of a transitive graph G generated by a TCK family S = (Sv,Se) on H. If S is algebraically isomorphic to LG, then S is analytic type.

Proof. By Theorem 9.6 there is an isomorphismϕ ˆ ∶ S → LG that maps the structure projection of S to the structure projection of LG, namely to 0. Hence, the structure projection of S is 0, so that S is weak-∗ homeomorphic completely isometrically isomorphic to LG via the canonical surjection as in Theorem 6.15, and S is analytic type.

10. Self-adjoint free semigroupoid algebras In spite of the nonself-adjoint nature of our algebras, a free semigroupoid algebra can occasionally turn out to be a von Neumann algebra. We have seen this phenomenon occur for cycle algebras, and Corollary 6.13 assures 50 K.R. DAVIDSON, A. DOR-ON, AND B. LI us that this can only occur when the supported graph for the TCK family is a disjoint union of transitive components. An example of Read [48] (c.f. [11]) shows that free semigroup algebras can also be self-adjoint, and this is our starting point.

Theorem 10.1 (Read). There exists isometries Z1,Z2 with pairwise orthogonal ranges, so that the free semigroup algebra Z generated by this family is B(H). + For any word γ = γ1⋯γn ∈ F2 of {1, 2}, we denote Zγ = Zγ1 Zγ2 ⋯Zγn . It is further shown in [11] that every rank one operator is a wot-limit of a + n bounded sequence of operators Tn ∈ span{Zγ ∶ γ ∈ F2 , SγS = 2 }. We ﬁrst show for any d ≥ 2, we can ﬁnd a family of Cuntz-isometries W1, ⋯,Wd, so that the free semigroup algebra generated by this family is B(H). This provides a correct argument for [11, Corollary 1.8].

Lemma 10.2. For every d ≥ 3, there exist Cuntz isometries W1, ⋯,Wd in B(H) so that the wot-closed algebra that they generate is B(H).

Proof. Let Z1,Z2 be a pair of Cuntz-isometries as in Read’s example. De- ﬁne isometries Wi by

W1 = Z1,W2 = Z21,...,Wd−1 = Z2⋯21, and Wd = Z2⋯22. d ∗ It is not hard to see that ∑i=1 WiWi = I. Let A be the algebra generated + by {W1,...,Wd}. For every word γ ∈ F2 of {1, 2}, we claim that Zγ1 = ZγZ1 belongs to A. This can be shown by induction on SγS. It is trivial when w is the empty word. If γ starts with the letter 1 and γ = 1 ⋅ γ′, then Zγ1 = W1Zγ 1 ∈ A by the induction. Otherwise, γ starts with letter 2. If ′ k γ = 2⋯21γ ,′ then we can factor Zγ1 = Wd WiZγ 1 ∈ A for some choice of k ≥ 0 k and 1 ≤ i ≤ d − 1. If γ = 22⋯2, then Zγ1 = Wd W′i ∈ A for some choice of k ≥ 0 and 1 ≤ i ≤ d − 1. n Therefore for every Tn ∈ span{Zγ ∶ SγS = 2 }, TnZ1 ∈ A. Since every rank ∗ one operator xy is a wot-limit of a sequence of such Tn, ∗ ∗ ∗ wot xy Z1 = x(Z1 y) ∈ A . ∗ wot However, Z1 is surjective, and thus every rank one operator lies in A . wot Since rank one operators are wot-dense in B(H), we have A = B(H). For free semigroupoid algebras of graphs that have more than one vertex, we would like to find examples that are self-adjoint. In the case of free semigroup algebras, which is just a free semigroupoid algebra where the graph has a single vertex, B(H) is the only self-adjoint example we know. We will construct a large class of directed graphs for which there exist free semigroupoid algebra that are self-adjoint. Definition 10.3. A directed graph G is called in-degree regular if the number of edges e ∈ E with r(e) = v is the same for every vertex v ∈ V . Similarly we define out-degree regular graphs. A transitive graph G is called aperiodic STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 51 if for any two vertices v, w ∈ V there exists a positive integer K0 so that for any K ≥ K0 there exists a directed path µ with SµS = K and s(µ) = v and r(µ) = w. Note that a transitive graph is p-periodic for p ≥ 2 if the vertices V can partitioned into disjoint sets V1,...,Vp so that s(e) ∈ Vi implies that r(e) ∈ V1+1(mod p). Our main goal of this section is to show that the free semigroupoid algebra associated with an aperiodic, in-degree regular, transitive, finite directed graph can be self-adjoint. The graph in Figure 10 is an example of such graph on three vertices with in-degree 2.

●

● ●

Figure 1. An aperiodic, in-degree regular, transitive, directed graph on 3 vertices

Definition 10.4. Let G = (V,E) be a directed graph. An edge colouring of G using d colours is a function c ∶ E → {1, 2, ⋯, d}. A colouring c is called a strong edge colouring if for every vertex v ∈ V , distinct edges with range v have different colours. Note that if G has a strong edge colouring with d colours, there can be at most d edges with range v. Moreover, if G is in-degree regular with in-degree d, this implies that for every colour 1 ≤ j ≤ d, there exists a unique edge e with r(e) = v and c(e) = j. An edge colouring c of the directed graph induces a colouring of paths + + c ∶ F (G) → Fd , where a directed path µ = ek . . . e1 is given the colour + c(µ) = c(ek) . . . c(e1) in the free semigroup Fd . Definition 10.5. Let G be a finite graph with an edge colouring c ∶ E → + {1, 2, . . . , d}. A word of colours γ = ck . . . c1 ∈ Fd is called a synchronizing word for a vertex v ∈ V if for any vertex w ∈ V (including v itself), there exists a path µ with s(µ) = v and r(µ) = w and c(µ) = γ. Remark 10.6. In the graph theory literature, it is standard (and more natural) to reverse the direction of our thinking and consider out-degree regular graphs. Then in a strong edge colouring, each vertex will have d edges with source v with distinct colours. A word of colours then yields a unique path of that colour with source v. A synchronizing word is then a word in colours with the property that every path with that word of colours terminates at the same vertex, independent of the initial vertex. The nomenclature synchronizing word perhaps makes sense more in this 52 K.R. DAVIDSON, A. DOR-ON, AND B. LI context. It can be interpreted as saying that there are universal directions to get to a vertex v by following the path of colour γ independent of the starting point w. Lemma 10.7. Fix a transitive, in-degree d-regular directed graph G with a strong edge colouring c by d colours. If there exists a synchronizing word γ for a vertex v ∈ V , then there exists a synchronizing word for any vertex w ∈ V . Proof. For any vertex w ∈ V , there is a directed path µ′ with s(µ′) = w and r(µ′) = v. Let the colour of the path µ′ be γ′ = c(µ′). We claim that γγ′ is a synchronizing word for the vertex w. Indeed, for any vertex u ∈ V , there exists a path µ with s(µ) = v, r(µ) = u, and c(µ) = γ. Since s(µ) = v = r(µ′), we can concatenate µ with µ′ to get s(µµ′) = w, r(µµ′) = u and c(µµ′) = γγ′. Therefore, γγ′ is a synchronizing word for the vertex w. Thus, for transitive in-degree regular graphs we can speak of a colouring being synchronizing without specifying the vertex. In this case, it is easy to see that the existence of a synchronized colouring implies aperiodicity. The converse, known as the road colouring problem, was a highly coveted result that was eventually established by Trahtman [49]. Theorem 10.8 (Trahtman). Let G be an aperiodic in-degree d regular transitive finite graph. Then G admits a strong edge colouring with d-colours that has a synchronizing word. A strong edge colouring enables us to define a CK family. Let G be in-degree regular with a strong edge colouring c. Let W1, ⋯,Wd be Cuntz ∗ isometries on B(H), i.e. Wi Wi = I for 1 ≤ i ≤ d and d ∗ Q WjWj = I. j=1 ⊕ Define K = ∑v∈V Hv, where each Hv ≅ H via a fixed unitary Jv. For each vertex v ∈ V , define Sv = PHv , the projection onto Hv. For each edge ∗ e ∈ E, with s(e) = w and r(e) = v, define Se = Jv Wc(e)Jw. The family Sc = (Sv,Se) is the CK family associated with the strong edge colouring c and the isometries W1,...,Wd. Indeed, it is not hard to verify Sc is a CK family. For each edge e, ∗ ∗ Se Se = JwJw = Sw = Ss(e) ≠ 0. and, because the d distinct edges with r(e) = v have distinct colours, d ∗ ∗ ∗ ∗ ∗ ∗ Q SeSe = Q Jv Wc(e)Wc(e)Jv = Jv Q WiWi Jv = Jv Jv = Sv. r(e)=v r(e)=v i=1

Therefore, Sc is a CK family. Let us ﬁx an aperiodic, in-degree regular, transitive, ﬁnite directed graph G of in-degree d. Use Trahtman’s Theorem 10.8 to select a strong edge STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 53 colouring which is synchronizing. Let us also use Lemma 10.2 to choose the isometries W1, ⋯,Wd ∈ B(H) so that the free semigroup algebra that they generate is B(H). For each word γ = cm . . . c1 of colours, denote Wγ =

Wcm ...Wc1 . Let Sc be the free semigroupoid algebra that Sc generates. We now prove a few lemmas towards the main result. Lemma 10.9. Let c be a strong edge colouring of an in-degree regular directed graph G. For any vertex v ∈ V and any word γ of colours, there exists a unique path µ with r(µ) = v and c(µ) = γ.

Proof. Write γ = cmcm−1 . . . c1. The lemma is trivial if m = 0. Since c is a strong edge colouring, there exists a unique edge em with c(em) = cm and r(em) = v. This reduces the length of γ by 1. By induction, we obtain a unique path µ = em . . . e1 in G such that c(ei) = ci and r(µ) = v. Lemma 10.10. Let c be a strong edge colouring of an in-degree regular directed graph G. Suppose there exists a synchronizing word γ for a vertex v ∈ V . Then Sc,v ∶= SvScSv = B(Hv). Proof. For each word γ′ of colours, Lemma 10.9 provides a unique word µ′ with r(µ′) = v and c(µ′) = γ′. Let w = s(µ′). Since γ is synchronizing, there exists a path µ with s(µ) = v, r(µ) = w and c(µ) = γ. Since r(µ) = w = ′ ′ s(µ ), the path λ = µ µ satisfies s(λ) = r(λ) = v. Therefore, Sλ belongs to ∗ ′ Sc,v = SvScSv. Notice that Sλ = Jv Wγ WγJv. Therefore for any word γ of ∗ colours, Jv Wγ WγJv ∈ B(Hv) belongs to′ Sc. wot ′ ′ + ∗ ′ Since Alg {Wγ ∶ γ ∈ Fd } = B(H), we see that Sc contains Jv A WγJv ′ ′ ∗ for any A ∈ B(Hv).′ In particular, taking A = AWγ for A ∈ B(H) shows that Sc,v = B(Hv). We can now complete the construction. Theorem 10.11. If G is an aperiodic, in-degree regular, transitive, finite directed graph, then there exists a strong edge colouring c so that the free semigroupoid algebra Sc is B(K). Proof. By Theorem 10.8, G has a strong edge colouring with a synchronizing word. It follows from the Lemma 10.7 that all vertices have synchronizing words. Hence by Lemma 10.10, SvScSv = B(Hv) for every vertex v ∈ V . Let v, w ∈ V be distinct vertices. There exists a path µ with s(µ) = w and r(µ) = v. Thus Sµ maps Hw isometrically into Hv. Since B(Hv) is contained in Sc, we see that ASµ lies in Sc for all A ∈ B(Hv). In particular, ∗ setting A = BSµ for B ∈ B(Hw, Hv) shows that B lies in Sc. Therefore, B(Hw, Hv) ⊆ Sc. We conclude that Sc = B(K). Example 10.12 (O’Brien [42]). Let G be an aperiodic, in-degree regular, transitive, finite graph that contains a loop (i.e. an edge e with r(e) = s(e) = v). We can explicitly construct a synchronizing word for this graph. We first arbitrarily colour the loop e at the vertex v with colour c1. Since G is transitive, we can find a directed spanning tree starting at v. We 54 K.R. DAVIDSON, A. DOR-ON, AND B. LI colour all edges in this spanning tree by c1 as well. Notice that for each vertex w ≠ v, there is precisely one edge in the spanning tree that points into w. Therefore, we can colour the remaining edges to obtain a strong edge colouring. Suppose the depth of the spanning tree is k. Then the word k γ = c1 is a synchronizing word. Indeed, for any w ∈ V , we can find a path µ along the spanning tree with s(µ) = v and r(µ) = w. Let the length of µ be m ≤ k, and set µ′ = µek−m be the path that begins with the loop k − m times followed by µ. Then the length of µ′ is k with s(µ′) = v and r(µ′) = w. All edges in µ′ are coloured ′ by c1. Therefore, c(µ ) = γ. Hence γ is a synchronizing word. When the graph is periodic, it is impossible to have a synchronizing word in the sense of the Definition 10.5 due to periodicity. However, there are some generalizations of Trahtman’s result to the periodic graphs [4] when one relaxes the definition of a synchronizing word. Question 10.13. If G is a transitive, periodic, in-degree d-regular finite graph with d ≥ 3, is there a free semigroupoid algebra on G which is von Neumann type? in particular B(H)? A technical question which we are unable to answer could resolve this: is there a family of Cuntz isometries W1, ⋯,Wd such that the wot-closed + algebra generated by {Wµ ∶ SµS = m, µ ∈ Fd } is B(H) for every m ≥ 1? We know from [11, Theorem 1.7] that the answer is affirmative when d = 2. As a result, we are able to extend Theorem 10.11 to the periodic case with in- degree 2. If the answer to this question is affirmative for all d ≥ 2, one would be able to extend Theorem 10.11 for every in-degree regular, transitive, finite directed graph. Example 10.14. The assumption of in-degree regular is essential in our construction as it allows us to convert our problem to edge colouring. How- ever, this assumption is certainly not necessary for the free semigroupoid algebra to be self-adjoint. Consider a graph G on n vertices v1, ⋯, vn. Let ei be edges with s(ei) = vi and r(ei) = vi+1 for all 1 ≤ i ≤ n−1. Let f1, f2 be two edges with s(fj) = vn and r(fj) = v1. This graph is a periodic, transitive directed graph that is not in-degree regular. Define S(ei) = I and S(fj) = Zj where Z1,Z2 are Cuntz isometries as in Read’s example. It is not hard to check that S is a CK family for G, and that the free semigroupoid algebra generated by S is B(K).

11. Atomic representations In this section, we classify a set of interesting examples of free semigroupoid algebras which generalize the atomic representations of the Cuntz- Toeplitz algebra studied in [19, Section 3]. This corresponds to the case where G consists of a single vertex with several edges. We shall prove a similar result for atomic free semigroupoid algebras. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 55

Deﬁnition 11.1. Let G be a directed graph. We call a TCK family S = (Sv,Se) atomic if there exists an orthonormal basis {ξv,i ∶ v ∈ V, i ∈ Λv} of H, sets Λv for v ∈ V and injections πe ∶ Λs(e) → Λr(e) such that

(1) For each vertex v ∈ V , Sv is the projection onto span{ξv,i ∶ i ∈ Λv}. (2) For each e ∈ E and i ∈ Λs(e), there are scalars λe,i ∈ T so that

Seξs(e),i = λe,iξr(e),πe(i), and Se maps all other basis vectors to 0. (3) The ranges of {πe ∶ e ∈ E} are pairwise disjoint in Λ = "v∈V Λv.

It is not hard to see that S = (Sv,Se) defines a TCK family for the directed graph G. Indeed, each Sv is a projection and their ranges are pairwise orthogonal. If e1, ⋯, ek have the same range w, then Sei have pairwise orthogonal ranges dominated by the range of Sw. One can consider atomic TCK families as a “combinatorially” defined TCK family. We shall prove a similar result to the one in [19, Section 3] showing that + the atomic representations of the free semigroupoid F (G) have three types: left regular type, inductive type, and cycle type. First of all, let us define a labeled directed graph H associated with the atomic representation. The vertices of H are precisely the set of basis vectors {ξv,j}. An edge points from a vertex ξv,i to ξu,j if there exists some e ∈ E with s(e) = v and i ∈ Λv so that r(e) = u and πe(i) = j, and the edge is labelled e. Equivalently, there’s an edge when Seξv,i = λξu,j for some λ ∈ T. For each vertex ξv,i in this graph, we let the number of edges pointing into this vertex be the in-degree, denoted by degin(ξv,i). Similarly, let the number of edges pointing out from this vertex be the out-degree, denoted by degout(ξv,j). We begin with a few simple observations.

Lemma 11.2. Let G be a directed graph and S = (Sv,Se) an atomic TCK family for G as in Deﬁnition 11.1. Suppose that H is the labeled directed graph associated with S. (1) Each connected component of H corresponds to a reducing subspace. (2) We have degin(ξv,i) ≤ 1 and degout(ξv,i) = degout(v) for each i ∈ Λv. ∗ (3) Sv − ∑e∈r 1(v) SeSe is the projection onto span{ξv,i ∶ degin(ξv,i) = 0}. − Proof. Identify each undirected connected component C of H with the span HC of the basis vectors which are vertices of C. This subspace is ∗ evidently invariant under each Se and Se for e ∈ E, and each basis vector is an eigenvector for every Sv. So HC is reducing for S. As the range of each Se corresponds to the range of πe, and these are pairwise disjoint, the in-degree of a vertex xv,i is either 1 or 0 depending on whether it is in the range of some πe. On the other hand, each e ∈ −1 s (v) determines a partial isometry with initial space span{ξv,i ∶ i ∈ Λv}. ∗ Therefore degout(ξv,i) = degout(v). Finally, the sot-sum ∑e∈r 1(v) SeSe has range consisting of −

span{ξv,i ∶ degin(ξv,i) = 1}.

The complement in Ran Sv is evidently span{ξv,i ∶ degin(ξv,i) = 0}. 56 K.R. DAVIDSON, A. DOR-ON, AND B. LI

The ﬁrst part of Lemma 11.2 tells us that we may restrict our attention to the case in which H is connected as an undirected graph, as the general case is a direct sum of such representations. The next lemma uses the fact that the in-degree is at most 1 to recursively select a unique predecessor of each vertex in the graph having in-degree 1.

Begin at some basis vector ξv0,i0 in H. If degin(ξv0,i0 ) = 0, the procedure stops. If degin(ξv0,i0 ) = 1, there is a unique vertex ξv 1,i 1 mapped to ξv0,i0 in

H by a unique edge e−1. Recursively deﬁne a sequence− ξ−v n,i n and edges e−n for n ≥ 0 by repeating this procedure. This backward path− either− terminates at a vector with in-degree 0, is an inﬁnite path with distinct vertices, or eventually repeats a vertex and hence becomes periodic. This establishes the following.

Proposition 11.3. Let G be a directed graph, and let S = (Sv,Se) an atomic TCK family for G. Suppose that H is the labeled directed graph associated with S. Let ξv0,i0 be a standard basis vector. Construct the sequence ξv n,i n and edges e−n as above. Then there are three possibilities: − −

(1) There is an n ≥ 0 so that degin(ξv n,i n ) = 0.

(2) The sequence ξv n,i n consists of distinct− − vertices.

(3) There is a smallest− − integer N and a smallest positive integer k ≥ 1

so that ξv n k,i n k = ξv n,i n and e−n−k = e−n for all n ≥ N.

− − − − − − Lemma 11.4. Let G be a directed graph, and let S = (Sv,Se) an atomic TCK family for G. Suppose that H is the labeled directed graph associated with S. Given a basis vector ξ = ξv,i, if there is no non-trivial path µ in H so that Sµξ = λξ, then ξ is a wandering vector. Moreover, we have the unitary S ≅ equivalence S S[ξv,i] LG,v. + Proof. Each Sµξ, for µ ∈ F (G) with s(µ) = v, is a non-zero multiple of another basis vector. Thus two such vectors are either orthogonal or scalar multiples. Suppose that there are two distinct paths µ, µ′ so that ′ Sµξ = λSµ ξ. Write µ = en . . . e1 and µ = fm . . . f1. With no loss of generality, we may suppose′ that n ≥ m. Because the in-degree of ξ is at most one, we see that en = fm. Therefore, we can inductively deduce that the last m edges ′ ′ ∗ in µ and µ coincide. It follows that µ = µ ν. Applying Sµ to the identity above, we obtain that Sνξ = λξ. Since this cannot happen′ non-trivially by hypothesis, ν must be the trivial path ν = v, and thus µ = µ′. Hence, ξ is a wandering vector. It follows that S[ξv,i] = span{Sµξv,i ∶ s(µ) = v}. It is evident from the action of S on this basis that the natural identiﬁcation U ∶ HG,v → S[ξv,i] given by Uξµ = Sµξv,i for s(µ) = v yields the desired unitary equivalence S between LG,v and S S[ξv,i].

Lemma 11.5. Let G be a directed graph, and let S = (Sv,Se) an atomic TCK family for G. Suppose that H is the labeled directed graph associated with S. If H is a connected graph, then it contains at most one cycle. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 57

Proof. Suppose that H has a cycle C and let H0 = H[C]. This is a directed subgraph of H in which every vertex has in-degree 1. As the in-degree in H is at most 1 at each vertex, it follows that H0 is a connected component of H. So H0 = H. Consider a vertex of the form ξv,i = Sµξ which is not in the cycle. Since the in-degree of each vertex is at most 1 and H = H[C], we can follow a path backward uniquely until it enters the cycle C. In particular, ξv,i cannot be in another cycle (from which there is no escape along a backward path). Therefore the connected component containing C has a unique cycle. The Wold decomposition theorem allows us to ﬁrst identify representations of left regular type. Proposition 11.6 (Left Regular Type). Let G be a directed graph, and let S be the free semigroupoid algebra generated by an atomic TCK family S = (Sv,Se) for G. Suppose that H is the labeled directed graph associated with S. If degin(ξv,i) = 0, then ξv,i is a wandering vector, and the connected component Cv,i of H containing ξv,i determines the reducing subspace Hv,i = S[ξv,i]. Moreover, the restriction of S to Hv,i is unitarily equivalent to LG,v. Additionally, if H is connected as an undirected graph and there is a vertex ξv,i with degin(ξv,i) = 0, then S is analytic. Next we consider the case of an inﬁnite tail (case (2) of Proposition 11.3). We synthesize the information in such a representation as follows.

Deﬁnition 11.7. Let G be a directed graph and τ = e−1e−2e−3 ... be an inﬁnite backward path in G. Let s(ei−1) = r(ei) = vi with i ≤ 0. For each vi for i ≤ 0, let Hi = HG,vi , and let Si = LG,vi act on Hi. Write Li,µ, for + µ ∈ F (G) for the elements of this family. There is a natural isometry

Ji ∶ Hi → Hi−1 given by Jiξµ = ξµei which satisfies Li−1,µJi = JiLi,µ. Let Hτ be the direct limit Hilbert space H = lim(H ,J ), which we can also consider τ Ð→ i i as the completion of the union of an increasing sequence of subspaces. Let Ki ∶ Hi → Hτ be the natural injections. Then we define Sµ on Hτ such that SµKi = KiLi,µ. This defines a TCK family Sτ . ⊕ ⊕ Since ∑e∈E SeHi = Hi ⊖ Cξvi , it follows that ∑e∈E SeHτ = Hτ . We deduce that Sτ is a non-degenerate fully coisometric CK family. In this case, we show that the free semigroupoid algebra is still analytic type.

Proposition 11.8. Let G be a directed graph, and let τ = e−1e−2e−3 ... be ′ an infinite backward path in G. Let G = G[{v−n ∶ n ≥ 0}] where vi = s(ei−1). Suppose Sτ is the free semigroupoid algebra of Sτ as in Definition 11.7. Then Sτ is completely isometrically isomorphic and weak-∗ homeomorphic to LG . ′ Proof. We adopt the same notation as in Definition 11.7. For each i ≤ 0, the restriction of Sτ to the invariant subspace Hi is unitarily equivalent to LG,vi . That Sτ is a wot-inductive and wot-projective limit of these algebras. 58 K.R. DAVIDSON, A. DOR-ON, AND B. LI

That Sτ isd completely isometrically isomorphic and weak-∗ homeomorphic to LG follows in exactly the same manner as Corollary 3.4. ′ The following result is now straightforward.

Proposition 11.9 (Inductive Type). Let G be a directed graph and τ = e−1e−2 ... , be an inﬁnite backward path in G. Let S = (Sv,Se) be an atomic TCK family for G that generates a free semigroupoid algebra S. Suppose that the labeled directed graph H associated with S is connected as an undirected graph and that the sequence (ξv n,i n ) in Proposition 11.3 consists of distinct vertices in H with connecting− labeled− edges ei. Then S is unitarily equivalent to Sτ , and S is completely isometrically isomorphic and weak-∗ ′ homeomorphic to LG , where G = G[{v−n ∶ n ≥ 0}]. ′ Corollary 11.10. Suppose G is a directed graph and S is an atomic TCK family for G. If S is a direct sum of inductive or regular types, then it is analytic.

Finally we consider the cycle type (case (3) in Proposition 11.3). Assume that H is connected and contains a cycle C. Let us denote the vertices of C as ξ1, . . . , ξk and e1, . . . , ek the labels of edges between them, and λ1, . . . , λk the scalars in T such that

Sej ξi = λjξj+1 for 1 ≤ j < k and Sek ξk = λkξ1.

Let vj be the vertices of G so that ξj = Svj ξj. Then for every e ≠ ej in E with s(e) = vj, the vector Seξj is a wandering vector by Lemma 11.4. We can visualize this setup as the cycle C from which various edges map each node ξj to basis vectors which do not lie in C. By Lemma 11.4, these vectors are wandering vectors which generate a copy of HG,v if they are in the range of Sv. Think of the cycle as having many copies of HG,v, for v ∈ V , attached. Note also that the edges ej ﬁt together, end to end, and form a cycle in G. We formalize this as a deﬁnition.

Deﬁnition 11.11. Suppose G is a directed graph, let w = ek . . . e1 be a cycle in G, and let λ ∈ T. Form a Hilbert space

k k ⊕ k Hw = C ⊕ Q Q HG,r(f) = C ⊕ Kw. j=1 f≠ej s(f)=s(ej )

k Choose an orthonormal basis ξ1, . . . , ξk for C , and for each 1 ≤ j ≤ k and each f ∈ E such that s(f) = s(ej) and f ≠ ej, denote the standard basis for a copy of HG,r(f) by ξj,f,µ for s(µ) = r(f). Deﬁne a fully coisometric CK family Sw,λ on Hw as follows: = = ∈ Svξj δv,s(ej )ξj and Svξj,f,µ δv,r(µ)ξj,f,µ for v V. STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 59

And for e ∈ E, deﬁne ⎪⎧ξ if e = e , 1 ≤ j < k ⎪ j+1 j ⎪λξ if e = e , j = k S ξ = ⎨ 1 k e j ⎪ ⎪ξj,e,r(e) if e ≠ ej, s(e) = s(ej) ⎪ ⎩⎪0 if s(e) ≠ s(ej) and ⎧ ⎪ξ if s(e) = r(µ) S ξ = ⎨ j,f,eµ e j,f,µ ⎪ ( ) ≠ ( ) ⎩⎪0 if s e s µ . Proposition 11.12 (Cycle Type). Let G be a directed graph, and let S = (Sv,Se) an atomic TCK family for G. Suppose that the labeled directed graph H associated with S is connected as an undirected graph and contains a cycle C with basis ξ1, . . . , ξk, edges e1, . . . , ek, and scalars λ1, . . . , λk in T such that

Sej ξi = λjξj+1 for 1 ≤ j < k and Sek ξk = λkξ1. k Define w = ek . . . e1 and λ = ∏j=1 λk. Then S is unitarily equivalent to Sw,λ. Proof. This proposition is established by rescaling the basis. First replace j−1 = ≤ < the basis vectors ξj by ∏i=1 λi ξj, so that now Sej ξj ξj+1 for 1 j k and Sek ξk = λξ1. Then for each f ∈ E with s(f) = s(ej) and f ≠ ej and + µ ∈ F (G) with s(µ) = r(f), we rescale so that each Sµf ξj is a basis vector rather than a scalar multiple of one. After this change, it is clear that we have a representation unitarily equivalent to Sw,λ. We note that the decomposition of any atomic representation into a direct sum of atomic representations on undirected connected components of H is canonical. However these subrepresentations are not always irreducible. So we now consider the decomposition of these representations into irreducible representations. This will provide an analog of [19, Proposition 3.10] for the case of free semigroup algebras. The proof itself follows the lines of the free semigroup case in [19] quite closely, and so we omit the details. + Definition 11.13. Let G be a directed graph. A cycle w = ek . . . e1 ∈ F (G) is called primitive if it is not a power of a strictly smaller word. Alternatively, w is not primitive when there is a word u and a positive integer p ≥ 2 so that w = up. ′ ′ ′ ′ Definition 11.14. Two infinite paths τ = e−1e−2e−3 ... and τ = e−1e−2e−3 ... ′ are shift tail equivalent if there are integers N and k so that em+k = em for all m ≤ N. An infinite path is eventually periodic if it is shift tail equivalent to a periodic path τ, in the sense that em−k = em for all m ≤ 0. If k is chosen to be minimal, so that the cycle u = e−1 . . . e−k is primitive, then τ has period k and we write π = u∞. Theorem 11.15. Let G be a directed graph. 60 K.R. DAVIDSON, A. DOR-ON, AND B. LI

(1) The left regular TCK family LG,v is irreducible for every v ∈ G. Two irreducible representations of left regular type are unitarily equivalent if and only if they arise from the same vertex. (2) Let w = ek . . . e1 be a cycle in G, and let λ ∈ T. Then Sw,λ is irreducible if and only if w is primitive. Two irreducible cycle representations Sw,λ and Su,µ are unitarily equivalent if and only if u is a cyclic permutation of w and µ = λ. If w = up for some primitive cycle u, then letting θj denote the pth roots of λ, we have a decomposition into irreducible families p

Sw,λ ≅ Q ⊕Su,θj . j=1

(3) Let τ = e−1e−2e−3 ... be an infinite backward path. Two inductive type atomic representations are unitarily equivalent if and only if the two infinite words are shift tail equivalent. Moreover, τ is not eventually periodic if and only if Sτ is irreducible, and when it is eventually periodic, say τ is shift tail equivalent to u∞ for a primitive cycle u = e−1 . . . e−k, then ⊕ Sτ ≅ S Su,λ dλ. T Two inductive type atomic representations are unitarily equivalent if and only if the two infinite words are shift tail equivalent. Remark 11.16. To decide if two atomic representations are unitarily equivalent, first decompose the graph H into connected components, which induces a decomposition of the representations into a direct sum of left regular, infinite tail and cycle types. Then split each cycle representation into irreducible ones. Next, count the multiplicities of each representation to decide unitary equivalence. It is not necessary to decompose periodic infinite tails into a direct integral as with two cycle representations, an infinite tail representation up to shift-tail equivalence is either appears, or does not appear in the direct sum decomposition. In fact, the direct integral over a proper subset A ⊂ T does not yield an atomic representation.

Finally, we explain the structure of Sw,λ for the cycle type representations.

Proposition 11.17. Let G be a directed graph, w = ek . . . e1 a primitive cycle in G and λ ∈ T. Suppose that Sw,λ is given as in Deﬁnition 11.11. k Then with respect to the decomposition Hw = C ⊕ Kw, we obtain

Mk 0 Sw,λ ≅ k ⊕ (αv) B(C , Kw) L ∑v∈V LG,v k where αv = ∑j=1 T{f ∈ E ∶ s(f) = vj, r(f) = v and f ≠ ej}T; and the 2, 2 corner ′ is unitarily equivalent to LG where G = G[{v ∈ V ∶ αv ≠ 0}]. In fact, the ′ STRUCTURE OF FREE SEMIGROUPOID ALGEBRAS 61

k projection of Hw onto C is the structure projection for Sw,λ, and Sw,λ is of dilation type.

Proof. Notice that when w is primitive, λSwξ1 = ξ1. If we begin with any ξj for 2 ≤ j ≤ k, the fact that w does not coincide with the cyclic permutation of w which maps ξj to λξj means that λSwξj is a multiple of a basis vector which is not in the cycle. Hence by Lemma 11.4, every standard basis vector except for ξ1 is mapped by λSw to a wandering vector. Therefore higher powers of this word map each basis vector to a sequence of basis vectors converging weakly to 0. Consequently, k ∗ wot–lim(λSw) = ξ1ξ n→∞ 1 ∗ ∗ belongs to S. Thus so does ξjξ1 = Sej 1...e1 ξ1ξ1 for 1 ≤ j ≤ k. Similarly ∗ k ξjξi belongs to Sw,λ for 1 ≤ i, j ≤ k, and− these operators span Mk = B(C ). ∗ Each ξi is a cyclic vector for Sw,λ, and from this we can deduce that ξj,f,µξi k belongs to Sw,λ; whence Sw,λ contains B(C , Kw). Also Kw is invariant, so the 1, 2 entry of the operator matrix is 0. So now we may consider the restriction of Sw,λ to Kw. Each ξj,f,r(f) is a wandering vector, and it generates a subspace canonically identiﬁed with HG,r(f). From the Wold decomposition Theorem 4.2, the restriction of Sw,λ to Kw is a direct sum of copies of LG,v where the multiplicities are determined by the dimension of the corresponding wandering space. These multiplicities are precisely the cardinalities of the sets of ξj,f,r(f) for which r(f) = v, f ≠ ej and s(f) = v for some j. That is, the multiplicity at v is exactly αv. Finally it follows from Theorem 3.1 that the free semigroupoid algebra on Kw is completely isometrically isomorphic and weak-∗ homeomorphic to ′ LG where G = G[{v ∈ V ∶ αv ≠ 0}]. The rest then follows from the structure theorem′ 6.15.

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Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]